Problem 4201
| ODE |
\[ \boxed {x \left (x -2 y\right ) {y^{\prime }}^{2}-2 y y^{\prime } x -2 y x +y^{2}=0} \] |
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program solution |
\[ y = -i x \] Verified OK. \[ y = i x \] Verified OK. \[ y = 0 \] Verified OK. \[ y = -i x \] Verified OK. \[ y = i x \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {2 \textit {\_a}^{2}+\sqrt {2}\, \sqrt {\textit {\_a} \left (\textit {\_a} -1\right )^{2}}}{\textit {\_a} \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} \right )+2 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {\sqrt {2}\, \sqrt {\textit {\_a} \left (\textit {\_a} -1\right )^{2}}-2 \textit {\_a}^{2}}{\textit {\_a} \left (\textit {\_a}^{2}+1\right )}d \textit {\_a} +2 c_{1} \right ) x \\ \end{align*} |
Problem 4202
| ODE |
\[ \boxed {x \left (x -2 y\right ) {y^{\prime }}^{2}+6 y y^{\prime } x -2 y x +y^{2}=0} \] |
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program solution |
\[ y = -x \sqrt {3}+2 x \] Verified OK. \[ y = x \sqrt {3}+2 x \] Verified OK. \[ x = \frac {c_{2} {\left (\frac {x \left (x -2 y\right ) \sqrt {\frac {-4 \left (x -\frac {y}{2}\right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 y x^{2}-7 x y^{2}+2 y^{3}}{x \left (x -2 y\right )^{2}}}-2 x^{2}+y x +\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}}{x \left (x -2 y\right )}\right )}^{\frac {\sqrt {\frac {\left (-\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x \left (y+x \right )\right )^{2} \left (-4 \left (x -\frac {y}{2}\right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 y x^{2}-7 x y^{2}+2 y^{3}\right )}{x^{3} \left (x -2 y\right )^{4}}}\, x \left (x -2 y\right )}{\sqrt {\frac {-4 \left (x -\frac {y}{2}\right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 y x^{2}-7 x y^{2}+2 y^{3}}{x \left (x -2 y\right )^{2}}}\, \left (-\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x \left (y+x \right )\right )}}}{\sqrt {\frac {-4 \left (x -\frac {y}{2}\right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 y x^{2}-7 x y^{2}+2 y^{3}}{x \left (x -2 y\right )^{2}}}} \] Warning, solution could not be verified \[ x = \frac {c_{2} {\left (\frac {x \left (x -2 y\right ) \sqrt {\frac {4 \left (x -\frac {y}{2}\right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 y x^{2}-7 x y^{2}+2 y^{3}}{x \left (x -2 y\right )^{2}}}-2 x^{2}+y x -\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}}{x \left (x -2 y\right )}\right )}^{\frac {\sqrt {\frac {\left (\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x \left (y+x \right )\right )^{2} \left (4 \left (x -\frac {y}{2}\right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 y x^{2}-7 x y^{2}+2 y^{3}\right )}{x^{3} \left (x -2 y\right )^{4}}}\, x \left (x -2 y\right )}{\left (\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{2}+y x \right ) \sqrt {\frac {4 \left (x -\frac {y}{2}\right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 y x^{2}-7 x y^{2}+2 y^{3}}{x \left (x -2 y\right )^{2}}}}}}{\sqrt {\frac {4 \left (x -\frac {y}{2}\right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 y x^{2}-7 x y^{2}+2 y^{3}}{x \left (x -2 y\right )^{2}}}} \] Warning, solution could not be verified \[ y = 0 \] Verified OK. \[ y = -x \sqrt {3}+2 x \] Verified OK. \[ y = x \sqrt {3}+2 x \] Verified OK. \[ x = \frac {c_{4} {\left (\frac {x \left (x -2 y\right ) \sqrt {\frac {-4 \left (x -\frac {y}{2}\right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 y x^{2}-7 x y^{2}+2 y^{3}}{x \left (x -2 y\right )^{2}}}-2 x^{2}+y x +\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}}{x \left (x -2 y\right )}\right )}^{-\frac {\sqrt {\frac {\left (-\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x \left (y+x \right )\right )^{2} \left (-4 \left (x -\frac {y}{2}\right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 y x^{2}-7 x y^{2}+2 y^{3}\right )}{x^{3} \left (x -2 y\right )^{4}}}\, x \left (x -2 y\right )}{\sqrt {\frac {-4 \left (x -\frac {y}{2}\right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 y x^{2}-7 x y^{2}+2 y^{3}}{x \left (x -2 y\right )^{2}}}\, \left (-\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x \left (y+x \right )\right )}}}{\sqrt {\frac {-4 \left (x -\frac {y}{2}\right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 y x^{2}-7 x y^{2}+2 y^{3}}{x \left (x -2 y\right )^{2}}}} \] Warning, solution could not be verified \[ x = \frac {c_{4} {\left (\frac {x \left (x -2 y\right ) \sqrt {\frac {4 \left (x -\frac {y}{2}\right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 y x^{2}-7 x y^{2}+2 y^{3}}{x \left (x -2 y\right )^{2}}}-2 x^{2}+y x -\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}}{x \left (x -2 y\right )}\right )}^{-\frac {\sqrt {\frac {\left (\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x \left (y+x \right )\right )^{2} \left (4 \left (x -\frac {y}{2}\right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 y x^{2}-7 x y^{2}+2 y^{3}\right )}{x^{3} \left (x -2 y\right )^{4}}}\, x \left (x -2 y\right )}{\left (\sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{2}+y x \right ) \sqrt {\frac {4 \left (x -\frac {y}{2}\right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 y x^{2}-7 x y^{2}+2 y^{3}}{x \left (x -2 y\right )^{2}}}}}}{\sqrt {\frac {4 \left (x -\frac {y}{2}\right ) \sqrt {2}\, \sqrt {x y \left (y+x \right )^{2}}+x^{3}+10 y x^{2}-7 x y^{2}+2 y^{3}}{x \left (x -2 y\right )^{2}}}} \] Warning, solution could not be verified |
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Maple solution |
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {2 \textit {\_a}^{2}+\sqrt {2}\, \sqrt {\textit {\_a} \left (\textit {\_a} +1\right )^{2}}-4 \textit {\_a}}{\textit {\_a} \left (\textit {\_a}^{2}-4 \textit {\_a} +1\right )}d \textit {\_a} \right )+2 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {\sqrt {2}\, \sqrt {\textit {\_a} \left (\textit {\_a} +1\right )^{2}}-2 \textit {\_a}^{2}+4 \textit {\_a}}{\textit {\_a} \left (\textit {\_a}^{2}-4 \textit {\_a} +1\right )}d \textit {\_a} +2 c_{1} \right ) x \\ \end{align*} |
Problem 4203
| ODE |
\[ \boxed {y^{2} {y^{\prime }}^{2}=a^{2}} \] |
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program solution |
\[ y = \sqrt {-2 a c_{2} -2 a x} \] Verified OK. \[ y = -\sqrt {-2 a c_{2} -2 a x} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \sqrt {2 a x +c_{1}} \\ y \left (x \right ) &= -\sqrt {2 a x +c_{1}} \\ y \left (x \right ) &= \sqrt {-2 a x +c_{1}} \\ y \left (x \right ) &= -\sqrt {-2 a x +c_{1}} \\ \end{align*} |
Problem 4204
| ODE |
\[ \boxed {y^{2} {y^{\prime }}^{2}+y^{2}=a^{2}} \] |
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program solution |
\[ -\sqrt {a^{2}-y^{2}} = x +c_{1} \] Verified OK. \[ \sqrt {a^{2}-y^{2}} = x +c_{2} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -a \\ y \left (x \right ) &= a \\ y \left (x \right ) &= \sqrt {a^{2}-c_{1}^{2}+2 c_{1} x -x^{2}} \\ y \left (x \right ) &= -\sqrt {\left (a +x -c_{1} \right ) \left (c_{1} +a -x \right )} \\ \end{align*} |
Problem 4205
| ODE |
\[ \boxed {y^{2} {y^{\prime }}^{2}-3 x y^{\prime }+y=0} \] |
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program solution |
\[ \ln \left (y\right )+\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {9 x^{2}-4 y^{3}}}{3 x}\right )}{3} = c_{1} \] Verified OK. \[ \ln \left (y\right )-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {9 x^{2}-4 y^{3}}}{3 x}\right )}{3} = c_{1} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {18^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{2} \\ y \left (x \right ) &= -\frac {2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}} \left (3 i 3^{\frac {1}{6}}+3^{\frac {2}{3}}\right )}{4} \\ y \left (x \right ) &= \frac {2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}} \left (3 i 3^{\frac {1}{6}}-3^{\frac {2}{3}}\right )}{4} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-3 \left (\int _{}^{\textit {\_Z}}\frac {4 \textit {\_a}^{3}+3 \sqrt {-4 \textit {\_a}^{3}+9}-9}{\textit {\_a} \left (4 \textit {\_a}^{3}-9\right )}d \textit {\_a} \right )+2 c_{1} \right ) x^{\frac {2}{3}} \\ \end{align*} |
Problem 4206
| ODE |
\[ \boxed {y^{2} {y^{\prime }}^{2}-6 x^{3} y^{\prime }+4 y x^{2}=0} \] |
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program solution |
\[ \frac {3 \ln \left (y\right )}{4}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {9 x^{4}-4 y^{3}}}{3 x^{2}}\right )}{2} = c_{1} \] Verified OK. \[ \frac {3 \ln \left (y\right )}{4}-\frac {\operatorname {arctanh}\left (\frac {\sqrt {9 x^{4}-4 y^{3}}}{3 x^{2}}\right )}{2} = c_{1} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {18^{\frac {1}{3}} x^{\frac {4}{3}}}{2} \\ y \left (x \right ) &= -\frac {18^{\frac {1}{3}} x^{\frac {4}{3}} \left (1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {18^{\frac {1}{3}} x^{\frac {4}{3}} \left (i \sqrt {3}-1\right )}{4} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-4 \ln \left (x \right )-3 \left (\int _{}^{\textit {\_Z}}\frac {4 \textit {\_a}^{3}+3 \sqrt {-4 \textit {\_a}^{3}+9}-9}{\textit {\_a} \left (4 \textit {\_a}^{3}-9\right )}d \textit {\_a} \right )+4 c_{1} \right ) x^{\frac {4}{3}} \\ \end{align*} |
Problem 4207
| ODE |
\[ \boxed {y^{2} {y^{\prime }}^{2}-4 y y^{\prime } a +y^{2}=-4 a^{2}+4 a x} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= -2 \sqrt {a x} \\ y \left (x \right ) &= 2 \sqrt {a x} \\ y \left (x \right ) &= \sqrt {4 a x -c_{1}^{2}+2 c_{1} x -x^{2}} \\ y \left (x \right ) &= -\sqrt {-x^{2}+\left (4 a +2 c_{1} \right ) x -c_{1}^{2}} \\ \end{align*} |
Problem 4208
| ODE |
\[ \boxed {y^{2} {y^{\prime }}^{2}-\left (x +1\right ) y y^{\prime }=-x} \] |
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program solution |
\[ y = \sqrt {x^{2}+2 c_{2}} \] Verified OK. \[ y = -\sqrt {x^{2}+2 c_{2}} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \sqrt {2 x +c_{1}} \\ y \left (x \right ) &= -\sqrt {2 x +c_{1}} \\ y \left (x \right ) &= \sqrt {x^{2}+c_{1}} \\ y \left (x \right ) &= -\sqrt {x^{2}+c_{1}} \\ \end{align*} |
Problem 4209
| ODE |
\[ \boxed {y^{2} {y^{\prime }}^{2}+2 y y^{\prime } x=-x^{2}} \] |
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program solution |
\[ y = \sqrt {-x^{2}+2 c_{1}} \] Verified OK. \[ y = -\sqrt {-x^{2}+2 c_{1}} \] Verified OK. \[ y = \sqrt {-x^{2}+2 c_{2}} \] Verified OK. \[ y = -\sqrt {-x^{2}+2 c_{2}} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \sqrt {-x^{2}+2 c_{1}} \\ y \left (x \right ) &= -\sqrt {-x^{2}+2 c_{1}} \\ \end{align*} |
Problem 4210
| ODE |
\[ \boxed {y^{2} {y^{\prime }}^{2}+2 y y^{\prime } x -y^{2}=-a} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= \sqrt {-x^{2}+a} \\ y \left (x \right ) &= -\sqrt {-x^{2}+a} \\ y \left (x \right ) &= \sqrt {c_{1}^{2}-2 c_{1} x +a} \\ y \left (x \right ) &= -\sqrt {c_{1}^{2}-2 c_{1} x +a} \\ \end{align*} |
Problem 4211
| ODE |
\[ \boxed {y^{2} {y^{\prime }}^{2}-2 y y^{\prime } x +2 y^{2}=x^{2}} \] |
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program solution |
\[ y = x \] Verified OK. \[ y = -i x \] Verified OK. \[ y = i x \] Verified OK. \[ y = -x \] Verified OK. \[ y = -i x \] Verified OK. \[ y = i x \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -x \\ y \left (x \right ) &= x \\ y \left (x \right ) &= \sqrt {-2 c_{1} \sqrt {2}\, x -c_{1}^{2}-x^{2}} \\ y \left (x \right ) &= \sqrt {2 c_{1} \sqrt {2}\, x -c_{1}^{2}-x^{2}} \\ y \left (x \right ) &= -\sqrt {-2 c_{1} \sqrt {2}\, x -c_{1}^{2}-x^{2}} \\ y \left (x \right ) &= -\sqrt {2 c_{1} \sqrt {2}\, x -c_{1}^{2}-x^{2}} \\ \end{align*} |
Problem 4212
| ODE |
\[ \boxed {y^{2} {y^{\prime }}^{2}-2 y y^{\prime } x +2 y^{2}=x^{2}-a} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {\sqrt {4 x^{2}-2 a}}{2} \\ y \left (x \right ) &= \frac {\sqrt {4 x^{2}-2 a}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {-8 c_{1}^{2}+16 c_{1} x -4 x^{2}-2 a}}{2} \\ y \left (x \right ) &= \frac {\sqrt {-8 c_{1}^{2}+16 c_{1} x -4 x^{2}-2 a}}{2} \\ \end{align*} |
Problem 4213
| ODE |
\[ \boxed {y^{2} {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (1-a \right ) y^{2}=-\left (a -1\right ) b -a \,x^{2}} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= \sqrt {-a \,x^{2}+b} \\ y \left (x \right ) &= -\sqrt {-a \,x^{2}+b} \\ y \left (x \right ) &= \sqrt {a \,c_{1}^{2}-2 a c_{1} x -c_{1}^{2}+2 c_{1} x -x^{2}+b} \\ y \left (x \right ) &= -\sqrt {\left (a -1\right ) c_{1}^{2}-2 x \left (a -1\right ) c_{1} -x^{2}+b} \\ \end{align*} |
Problem 4214
| ODE |
\[ \boxed {\left (1-y^{2}\right ) {y^{\prime }}^{2}=1} \] |
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program solution |
\[ -\frac {y \sqrt {1-y^{2}}}{2}-\frac {\arcsin \left (y\right )}{2} = x +c_{1} \] Verified OK. \[ \frac {y \sqrt {1-y^{2}}}{2}+\frac {\arcsin \left (y\right )}{2} = x +c_{2} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \sin \left (\operatorname {RootOf}\left (\sin \left (\textit {\_Z} \right ) \operatorname {csgn}\left (\cos \left (\textit {\_Z} \right )\right ) \cos \left (\textit {\_Z} \right )+\textit {\_Z} +2 c_{1} -2 x \right )\right ) \\ y \left (x \right ) &= \sin \left (\operatorname {RootOf}\left (-\sin \left (\textit {\_Z} \right ) \operatorname {csgn}\left (\cos \left (\textit {\_Z} \right )\right ) \cos \left (\textit {\_Z} \right )-\textit {\_Z} +2 c_{1} -2 x \right )\right ) \\ \end{align*} |
Problem 4215
| ODE |
\[ \boxed {\left (a^{2}-y^{2}\right ) {y^{\prime }}^{2}-y^{2}=0} \] |
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program solution |
\[ \sqrt {a^{2}-y^{2}}-\frac {a \ln \left (\sqrt {a^{2}-y^{2}}+a \right )}{2}+\frac {a \ln \left (\sqrt {a^{2}-y^{2}}-a \right )}{2} = x +c_{1} \] Verified OK. \[ -\sqrt {a^{2}-y^{2}}+\frac {a \ln \left (\sqrt {a^{2}-y^{2}}+a \right )}{2}-\frac {a \ln \left (\sqrt {a^{2}-y^{2}}-a \right )}{2} = x +c_{2} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= 0 \\ a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )+a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\operatorname {csgn}\left (a \right ) \sqrt {a^{2}-y \left (x \right )^{2}}+a \right )}{y \left (x \right )}\right )-\sqrt {a^{2}-y \left (x \right )^{2}}-c_{1} +x &= 0 \\ -a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )-a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\operatorname {csgn}\left (a \right ) \sqrt {a^{2}-y \left (x \right )^{2}}+a \right )}{y \left (x \right )}\right )+\sqrt {a^{2}-y \left (x \right )^{2}}-c_{1} +x &= 0 \\ \end{align*} |
Problem 4216
| ODE |
\[ \boxed {\left (a^{2}-2 x a y+y^{2}\right ) {y^{\prime }}^{2}+2 y y^{\prime } a +y^{2}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 4217
| ODE |
\[ \boxed {\left (\left (1-a \right ) x^{2}+y^{2}\right ) {y^{\prime }}^{2}+2 a x y y^{\prime }+\left (1-a \right ) y^{2}=-x^{2}} \] |
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program solution |
\[ y = -i x \] Verified OK. \[ y = i x \] Verified OK. \[ y = -i x \] Verified OK. \[ y = i x \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= \tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} \sqrt {a -1}-\ln \left (x^{2} \sec \left (\textit {\_Z} \right )^{2}\right )+2 c_{1} \right )\right ) x \\ y \left (x \right ) &= \tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} \sqrt {a -1}-\ln \left (x^{2} \sec \left (\textit {\_Z} \right )^{2}\right )+2 c_{1} \right )\right ) x \\ \end{align*} |
Problem 4218
| ODE |
\[ \boxed {\left (\left (-4 a^{2}+1\right ) x^{2}+y^{2}\right ) {y^{\prime }}^{2}-8 a^{2} x y y^{\prime }+\left (-4 a^{2}+1\right ) y^{2}=-x^{2}} \] |
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program solution |
\[ y = \frac {-x \,a^{2} \sqrt {8 a^{2}-1-4 \sqrt {4 a^{4}-a^{2}}}+x \sqrt {32 a^{6}-12 a^{4}-16 a^{4} \sqrt {4 a^{4}-a^{2}}+4 a^{2} \sqrt {4 a^{4}-a^{2}}+a^{2}}}{-a^{2}+\sqrt {4 a^{4}-a^{2}}} \] Verified OK. \[ y = \frac {x \,a^{2} \sqrt {8 a^{2}-1+4 \sqrt {4 a^{4}-a^{2}}}-x \sqrt {32 a^{6}-12 a^{4}+16 a^{4} \sqrt {4 a^{4}-a^{2}}-4 a^{2} \sqrt {4 a^{4}-a^{2}}+a^{2}}}{a^{2}+\sqrt {4 a^{4}-a^{2}}} \] Verified OK. \[ y = \frac {x \,a^{2} \sqrt {8 a^{2}-1-4 \sqrt {4 a^{4}-a^{2}}}+x \sqrt {32 a^{6}-12 a^{4}-16 a^{4} \sqrt {4 a^{4}-a^{2}}+4 a^{2} \sqrt {4 a^{4}-a^{2}}+a^{2}}}{-a^{2}+\sqrt {4 a^{4}-a^{2}}} \] Verified OK. \[ y = \frac {-x \,a^{2} \sqrt {8 a^{2}-1+4 \sqrt {4 a^{4}-a^{2}}}-x \sqrt {32 a^{6}-12 a^{4}+16 a^{4} \sqrt {4 a^{4}-a^{2}}-4 a^{2} \sqrt {4 a^{4}-a^{2}}+a^{2}}}{a^{2}+\sqrt {4 a^{4}-a^{2}}} \] Verified OK. \[ y = \frac {-x \,a^{2} \sqrt {8 a^{2}-1-4 \sqrt {4 a^{4}-a^{2}}}-x \sqrt {32 a^{6}-12 a^{4}-16 a^{4} \sqrt {4 a^{4}-a^{2}}+4 a^{2} \sqrt {4 a^{4}-a^{2}}+a^{2}}}{-a^{2}+\sqrt {4 a^{4}-a^{2}}} \] Verified OK. \[ y = \frac {x \,a^{2} \sqrt {8 a^{2}-1+4 \sqrt {4 a^{4}-a^{2}}}+x \sqrt {32 a^{6}-12 a^{4}+16 a^{4} \sqrt {4 a^{4}-a^{2}}-4 a^{2} \sqrt {4 a^{4}-a^{2}}+a^{2}}}{a^{2}+\sqrt {4 a^{4}-a^{2}}} \] Verified OK. \[ y = \frac {x \,a^{2} \sqrt {8 a^{2}-1-4 \sqrt {4 a^{4}-a^{2}}}-x \sqrt {32 a^{6}-12 a^{4}-16 a^{4} \sqrt {4 a^{4}-a^{2}}+4 a^{2} \sqrt {4 a^{4}-a^{2}}+a^{2}}}{-a^{2}+\sqrt {4 a^{4}-a^{2}}} \] Verified OK. \[ y = \frac {-x \,a^{2} \sqrt {8 a^{2}-1+4 \sqrt {4 a^{4}-a^{2}}}+x \sqrt {32 a^{6}-12 a^{4}+16 a^{4} \sqrt {4 a^{4}-a^{2}}-4 a^{2} \sqrt {4 a^{4}-a^{2}}+a^{2}}}{a^{2}+\sqrt {4 a^{4}-a^{2}}} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{3}-8 \textit {\_a} \,a^{2}-\sqrt {\left (4 a^{2}-1\right ) \left (\textit {\_a}^{2}+1\right )^{2}}+\textit {\_a}}{\textit {\_a}^{4}-16 \textit {\_a}^{2} a^{2}+2 \textit {\_a}^{2}+1}d \textit {\_a} +c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{3}-8 \textit {\_a} \,a^{2}+\sqrt {\left (4 a^{2}-1\right ) \left (\textit {\_a}^{2}+1\right )^{2}}+\textit {\_a}}{\textit {\_a}^{4}-16 \textit {\_a}^{2} a^{2}+2 \textit {\_a}^{2}+1}d \textit {\_a} \right )+c_{1} \right ) x \\ \end{align*} |
Problem 4219
| ODE |
\[ \boxed {\left (\left (-a^{2}+1\right ) x^{2}+y^{2}\right ) {y^{\prime }}^{2}+2 a^{2} x y y^{\prime }+\left (-a^{2}+1\right ) y^{2}=-x^{2}} \] |
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program solution |
\[ y = -i x \] Verified OK. \[ y = i x \] Verified OK. \[ y = -i x \] Verified OK. \[ y = i x \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -i x \\ y \left (x \right ) &= i x \\ y \left (x \right ) &= \tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} \sqrt {a^{2}-1}-\ln \left (x^{2} \sec \left (\textit {\_Z} \right )^{2}\right )+2 c_{1} \right )\right ) x \\ y \left (x \right ) &= \tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} \sqrt {a^{2}-1}-\ln \left (x^{2} \sec \left (\textit {\_Z} \right )^{2}\right )+2 c_{1} \right )\right ) x \\ \end{align*} |
Problem 4220
| ODE |
\[ \boxed {\left (y+x \right )^{2} {y^{\prime }}^{2}-y^{2}=0} \] |
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program solution |
\[ -\frac {x}{y}+\ln \left (\frac {y}{x}\right )+\ln \left (x \right )-c_{5} = 0 \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {x}{\operatorname {LambertW}\left (x \,{\mathrm e}^{c_{1}}\right )} \\ y \left (x \right ) &= -x -\sqrt {x^{2}+2 c_{1}} \\ y \left (x \right ) &= -x +\sqrt {x^{2}+2 c_{1}} \\ \end{align*} |
Problem 4221
| ODE |
\[ \boxed {\left (y+x \right )^{2} {y^{\prime }}^{2}-\left (x^{2}-y x -2 y^{2}\right ) y^{\prime }-\left (-y+x \right ) y=0} \] |
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program solution |
\[ \sqrt {\frac {y \left (2 x +y\right )}{x^{2}}} = \frac {c_{6} {\mathrm e}^{c_{5}}}{x} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -x -\sqrt {x^{2}+2 c_{1}} \\ y \left (x \right ) &= -x +\sqrt {x^{2}+2 c_{1}} \\ y \left (x \right ) &= \frac {-c_{1} x -\sqrt {2 c_{1}^{2} x^{2}+1}}{c_{1}} \\ y \left (x \right ) &= \frac {-c_{1} x +\sqrt {2 c_{1}^{2} x^{2}+1}}{c_{1}} \\ \end{align*} |
Problem 4222
| ODE |
\[ \boxed {\left (a^{2}-\left (-y+x \right )^{2}\right ) {y^{\prime }}^{2}+2 y^{\prime } a^{2}-\left (-y+x \right )^{2}=-a^{2}} \] |
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program solution |
\[ y = a \sqrt {2}+x \] Verified OK. \[ x = -\frac {\left (a^{2}-\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 y x -y^{2}\right )}\right ) a^{2} \sqrt {2}\, \left (-\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 y x -y^{2}\right )}+\left (-y+x \right )^{2}\right )}{2 \sqrt {-\frac {a^{4} \left (-a^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 y x -y^{2}\right )}\right ) \left (-x^{2}+2 y x -y^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 y x -y^{2}\right )}\right )^{2}}{\left (a^{2}-x^{2}+2 y x -y^{2}\right )^{4}}}\, \left (a +x -y\right )^{2} \left (a -x +y\right )^{2}}+c_{2} \] Verified OK. \[ x = -\frac {\left (a^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 y x -y^{2}\right )}\right ) \left (\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 y x -y^{2}\right )}+\left (-y+x \right )^{2}\right ) a^{2} \sqrt {2}}{2 \sqrt {\frac {a^{4} \left (a^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 y x -y^{2}\right )}\right ) \left (x^{2}-2 y x +y^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 y x -y^{2}\right )}\right )^{2}}{\left (a^{2}-x^{2}+2 y x -y^{2}\right )^{4}}}\, \left (a +x -y\right )^{2} \left (a -x +y\right )^{2}}+c_{2} \] Verified OK. \[ y = -a \sqrt {2}+x \] Verified OK. \[ x = \frac {\left (a^{2}-\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 y x -y^{2}\right )}\right ) a^{2} \sqrt {2}\, \left (-\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 y x -y^{2}\right )}+\left (-y+x \right )^{2}\right )}{2 \sqrt {-\frac {a^{4} \left (-a^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 y x -y^{2}\right )}\right ) \left (-x^{2}+2 y x -y^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 y x -y^{2}\right )}\right )^{2}}{\left (a^{2}-x^{2}+2 y x -y^{2}\right )^{4}}}\, \left (a +x -y\right )^{2} \left (a -x +y\right )^{2}}+c_{4} \] Verified OK. \[ x = \frac {\left (a^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 y x -y^{2}\right )}\right ) \left (\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 y x -y^{2}\right )}+\left (-y+x \right )^{2}\right ) a^{2} \sqrt {2}}{2 \sqrt {\frac {a^{4} \left (a^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 y x -y^{2}\right )}\right ) \left (x^{2}-2 y x +y^{2}+\sqrt {\left (-y+x \right )^{2} \left (2 a^{2}-x^{2}+2 y x -y^{2}\right )}\right )^{2}}{\left (a^{2}-x^{2}+2 y x -y^{2}\right )^{4}}}\, \left (a +x -y\right )^{2} \left (a -x +y\right )^{2}}+c_{4} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= x -\sqrt {2}\, a \\ y \left (x \right ) &= x +\sqrt {2}\, a \\ y \left (x \right ) &= x +\operatorname {RootOf}\left (-2 x -\left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2}-2 a^{2}+\sqrt {-\textit {\_a}^{4}+2 \textit {\_a}^{2} a^{2}}}{\textit {\_a}^{2}-2 a^{2}}d \textit {\_a} \right )+2 c_{1} \right ) \\ y \left (x \right ) &= x +\operatorname {RootOf}\left (-2 x +\int _{}^{\textit {\_Z}}-\frac {-2 a^{2}+\textit {\_a}^{2}-\sqrt {-\textit {\_a}^{4}+2 \textit {\_a}^{2} a^{2}}}{\textit {\_a}^{2}-2 a^{2}}d \textit {\_a} +2 c_{1} \right ) \\ \end{align*} |
Problem 4223
| ODE |
\[ \boxed {2 y^{2} {y^{\prime }}^{2}+2 y y^{\prime } x +y^{2}=-x^{2}+1} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {\sqrt {-2 x^{2}+4}}{2} \\ y \left (x \right ) &= \frac {\sqrt {-2 x^{2}+4}}{2} \\ y \left (x \right ) &= \sqrt {\operatorname {RootOf}\left (-2 \ln \left (x \right )+2 \,\operatorname {arctanh}\left (\sqrt {-2 \textit {\_Z} -1}\right )-\ln \left (\textit {\_Z} +1\right )+2 c_{1} \right ) x^{2}+1} \\ y \left (x \right ) &= -\sqrt {\operatorname {RootOf}\left (-2 \ln \left (x \right )+2 \,\operatorname {arctanh}\left (\sqrt {-2 \textit {\_Z} -1}\right )-\ln \left (\textit {\_Z} +1\right )+2 c_{1} \right ) x^{2}+1} \\ \end{align*} |
Problem 4224
| ODE |
\[ \boxed {3 y^{2} {y^{\prime }}^{2}-2 y y^{\prime } x +4 y^{2}=x^{2}} \] |
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program solution |
\[ y = -i x \] Verified OK. \[ y = i x \] Verified OK. \[ y = \frac {x \sqrt {3}}{3} \] Verified OK. \[ y = -i x \] Verified OK. \[ y = i x \] Verified OK. \[ y = -\frac {x \sqrt {3}}{3} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {3}\, x}{3} \\ y \left (x \right ) &= \frac {\sqrt {3}\, x}{3} \\ \ln \left (x \right )-\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-3 y \left (x \right )^{2}}{x^{2}}}}{2}\right )+\frac {\ln \left (\frac {x^{2}+y \left (x \right )^{2}}{x^{2}}\right )}{2}-c_{1} &= 0 \\ \ln \left (x \right )+\operatorname {arctanh}\left (\frac {\sqrt {\frac {x^{2}-3 y \left (x \right )^{2}}{x^{2}}}}{2}\right )+\frac {\ln \left (\frac {x^{2}+y \left (x \right )^{2}}{x^{2}}\right )}{2}-c_{1} &= 0 \\ \end{align*} |
Problem 4225
| ODE |
\[ \boxed {4 y^{2} {y^{\prime }}^{2}+2 \left (1+3 x \right ) x y y^{\prime }=-3 x^{3}} \] |
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program solution |
\[ y = \sqrt {-x^{3}+2 c_{2}} \] Verified OK. \[ y = -\sqrt {-x^{3}+2 c_{2}} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -\frac {\sqrt {-2 x^{2}+4 c_{1}}}{2} \\ y \left (x \right ) &= \frac {\sqrt {-2 x^{2}+4 c_{1}}}{2} \\ y \left (x \right ) &= \sqrt {-x^{3}+c_{1}} \\ y \left (x \right ) &= -\sqrt {-x^{3}+c_{1}} \\ \end{align*} |
Problem 4226
| ODE |
\[ \boxed {\left (x^{2}-4 y^{2}\right ) {y^{\prime }}^{2}+6 y y^{\prime } x +y^{2}=4 x^{2}} \] |
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program solution |
\[ \ln \left (1-\frac {y^{2}}{x^{2}}\right )+\operatorname {arctanh}\left (\frac {y}{x}\right )+2 \ln \left (x \right )-c_{5} = 0 \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {x \left (-\operatorname {RootOf}\left (\textit {\_Z}^{16}+2 \textit {\_Z}^{4} c_{1} x^{4}-c_{1} x^{4}\right )^{4}+1\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{16}+2 \textit {\_Z}^{4} c_{1} x^{4}-c_{1} x^{4}\right )^{4}} \\ y \left (x \right ) &= \frac {\frac {\operatorname {RootOf}\left (\textit {\_Z}^{16}-2 \textit {\_Z}^{4} c_{1} x^{4}-c_{1} x^{4}\right )^{12}}{c_{1}}-x^{4}}{x^{3}} \\ \end{align*} |
Problem 4227
| ODE |
\[ \boxed {9 y^{2} {y^{\prime }}^{2}-3 x y^{\prime }+y=0} \] |
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program solution |
\[ \ln \left (y\right )+\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{2}-4 y^{3}}}{x}\right )}{3} = c_{1} \] Verified OK. \[ \ln \left (y\right )-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {x^{2}-4 y^{3}}}{x}\right )}{3} = c_{1} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{2} \\ y \left (x \right ) &= -\frac {2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-3 \left (\int _{}^{\textit {\_Z}}\frac {4 \textit {\_a}^{3}+\sqrt {-4 \textit {\_a}^{3}+1}-1}{\textit {\_a} \left (4 \textit {\_a}^{3}-1\right )}d \textit {\_a} \right )+2 c_{1} \right ) x^{\frac {2}{3}} \\ \end{align*} |
Problem 4228
| ODE |
\[ \boxed {\left (2-3 y\right )^{2} {y^{\prime }}^{2}+4 y=4} \] |
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program solution |
\[ -y \sqrt {1-y} = x +c_{1} \] Verified OK. \[ y \sqrt {1-y} = x +c_{2} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= 1 \\ y \left (x \right ) &= -\frac {{\left (\left (108 c_{1} -108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}+\frac {12}{\left (108 c_{1} -108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )}^{2}}{36}+1 \\ y \left (x \right ) &= 1+\frac {{\left (\left (i-\sqrt {3}\right ) \left (108 c_{1} -108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}+12 i+12 \sqrt {3}\right )}^{2}}{144 \left (108 c_{1} -108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= 1+\frac {{\left (\left (\sqrt {3}+i\right ) \left (108 c_{1} -108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}+12 i-12 \sqrt {3}\right )}^{2}}{144 \left (108 c_{1} -108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {{\left (\left (-108 c_{1} +108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}+\frac {12}{\left (-108 c_{1} +108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {1}{3}}}\right )}^{2}}{36}+1 \\ y \left (x \right ) &= 1+\frac {{\left (\left (i-\sqrt {3}\right ) \left (-108 c_{1} +108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}+12 i+12 \sqrt {3}\right )}^{2}}{144 \left (-108 c_{1} +108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= 1+\frac {{\left (\left (\sqrt {3}+i\right ) \left (-108 c_{1} +108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}+12 i-12 \sqrt {3}\right )}^{2}}{144 \left (-108 c_{1} +108 x +12 \sqrt {81 c_{1}^{2}-162 c_{1} x +81 x^{2}-12}\right )^{\frac {2}{3}}} \\ \end{align*} |
Problem 4229
| ODE |
\[ \boxed {\left (-a^{2}+1\right ) y^{2} {y^{\prime }}^{2}-3 a^{2} x y y^{\prime }+y^{2}=a^{2} x^{2}} \] |
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program solution |
\[ y = \frac {-3 a^{2} x \sqrt {-\frac {2 \left (3 a^{2}+\sqrt {5 a^{4}-2 a^{2}+1}-1\right )}{a^{2}-1}}-a x \sqrt {-\frac {2 \left (15 a^{4}+5 \sqrt {5 a^{4}-2 a^{2}+1}\, a^{2}-a^{2}+4 \sqrt {5 a^{4}-2 a^{2}+1}+4\right )}{a^{2}-1}}}{6 a^{2}+2 \sqrt {5 a^{4}-2 a^{2}+1}+2} \] Verified OK. \[ y = \frac {3 a^{2} x \sqrt {-\frac {2 \left (3 a^{2}+\sqrt {5 a^{4}-2 a^{2}+1}-1\right )}{a^{2}-1}}-a x \sqrt {-\frac {2 \left (15 a^{4}+5 \sqrt {5 a^{4}-2 a^{2}+1}\, a^{2}-a^{2}+4 \sqrt {5 a^{4}-2 a^{2}+1}+4\right )}{a^{2}-1}}}{6 a^{2}+2 \sqrt {5 a^{4}-2 a^{2}+1}+2} \] Verified OK. \[ y = \frac {3 \sqrt {2}\, a^{2} x \sqrt {\frac {-3 a^{2}+1+\sqrt {5 a^{4}-2 a^{2}+1}}{a^{2}-1}}+\sqrt {2}\, a x \sqrt {\frac {-15 a^{4}+5 \sqrt {5 a^{4}-2 a^{2}+1}\, a^{2}+a^{2}+4 \sqrt {5 a^{4}-2 a^{2}+1}-4}{a^{2}-1}}}{-6 a^{2}+2 \sqrt {5 a^{4}-2 a^{2}+1}-2} \] Verified OK. \[ y = \frac {-3 \sqrt {2}\, a^{2} x \sqrt {\frac {-3 a^{2}+1+\sqrt {5 a^{4}-2 a^{2}+1}}{a^{2}-1}}+\sqrt {2}\, a x \sqrt {\frac {-15 a^{4}+5 \sqrt {5 a^{4}-2 a^{2}+1}\, a^{2}+a^{2}+4 \sqrt {5 a^{4}-2 a^{2}+1}-4}{a^{2}-1}}}{-6 a^{2}+2 \sqrt {5 a^{4}-2 a^{2}+1}-2} \] Verified OK. \[ y = \frac {-3 a^{2} x \sqrt {-\frac {2 \left (3 a^{2}+\sqrt {5 a^{4}-2 a^{2}+1}-1\right )}{a^{2}-1}}+a x \sqrt {-\frac {2 \left (15 a^{4}+5 \sqrt {5 a^{4}-2 a^{2}+1}\, a^{2}-a^{2}+4 \sqrt {5 a^{4}-2 a^{2}+1}+4\right )}{a^{2}-1}}}{6 a^{2}+2 \sqrt {5 a^{4}-2 a^{2}+1}+2} \] Verified OK. \[ y = \frac {3 a^{2} x \sqrt {-\frac {2 \left (3 a^{2}+\sqrt {5 a^{4}-2 a^{2}+1}-1\right )}{a^{2}-1}}+a x \sqrt {-\frac {2 \left (15 a^{4}+5 \sqrt {5 a^{4}-2 a^{2}+1}\, a^{2}-a^{2}+4 \sqrt {5 a^{4}-2 a^{2}+1}+4\right )}{a^{2}-1}}}{6 a^{2}+2 \sqrt {5 a^{4}-2 a^{2}+1}+2} \] Verified OK. \[ y = \frac {3 \sqrt {2}\, a^{2} x \sqrt {\frac {-3 a^{2}+1+\sqrt {5 a^{4}-2 a^{2}+1}}{a^{2}-1}}-\sqrt {2}\, a x \sqrt {\frac {-15 a^{4}+5 \sqrt {5 a^{4}-2 a^{2}+1}\, a^{2}+a^{2}+4 \sqrt {5 a^{4}-2 a^{2}+1}-4}{a^{2}-1}}}{-6 a^{2}+2 \sqrt {5 a^{4}-2 a^{2}+1}-2} \] Verified OK. \[ y = \frac {-3 \sqrt {2}\, a^{2} x \sqrt {\frac {-3 a^{2}+1+\sqrt {5 a^{4}-2 a^{2}+1}}{a^{2}-1}}-\sqrt {2}\, a x \sqrt {\frac {-15 a^{4}+5 \sqrt {5 a^{4}-2 a^{2}+1}\, a^{2}+a^{2}+4 \sqrt {5 a^{4}-2 a^{2}+1}-4}{a^{2}-1}}}{-6 a^{2}+2 \sqrt {5 a^{4}-2 a^{2}+1}-2} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {\left (2 \textit {\_a}^{2} a^{2}-2 \textit {\_a}^{2}+3 a^{2}+\sqrt {4 \textit {\_a}^{2} a^{2}+5 a^{4}-4 \textit {\_a}^{2}+4 a^{2}}\right ) \textit {\_a}}{a^{2} \textit {\_a}^{4}-\textit {\_a}^{4}+3 \textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}d \textit {\_a} \right )+2 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\left (2 \textit {\_a}^{2} a^{2}-2 \textit {\_a}^{2}+3 a^{2}-\sqrt {4 \textit {\_a}^{2} a^{2}+5 a^{4}-4 \textit {\_a}^{2}+4 a^{2}}\right ) \textit {\_a}}{a^{2} \textit {\_a}^{4}-\textit {\_a}^{4}+3 \textit {\_a}^{2} a^{2}-\textit {\_a}^{2}+a^{2}}d \textit {\_a} +2 c_{1} \right ) x \\ \end{align*} |
Problem 4230
| ODE |
\[ \boxed {\left (a -b \right ) y^{2} {y^{\prime }}^{2}-2 b x y y^{\prime }+y^{2} a=b \,x^{2}+a b} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {\sqrt {b \left (x^{2}+a -b \right ) \left (a -b \right )}}{a -b} \\ y \left (x \right ) &= -\frac {\sqrt {b \left (x^{2}+a -b \right ) \left (a -b \right )}}{a -b} \\ -\left (\int _{\textit {\_b}}^{x}\frac {b \textit {\_a} +\sqrt {a \left (\left (-a +b \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )\right )}}{\sqrt {a \left (\left (-a +b \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )\right )}\, \textit {\_a} +\left (-a +b \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )}d \textit {\_a} \right )+\int _{}^{y \left (x \right )}\frac {\left (\left (\sqrt {a \left (-b^{2}+\left (\textit {\_f}^{2}+x^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, x +\left (-a +b \right ) \textit {\_f}^{2}+b \left (x^{2}+a -b \right )\right ) \left (\int _{\textit {\_b}}^{x}\frac {\left (a -b \right ) \left (2 b \textit {\_a} \sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}+a \left (-b^{2}+\left (2 \textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )\right )}{\sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, {\left (\sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, \textit {\_a} -b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}^{2}}d \textit {\_a} \right )+a -b \right ) \textit {\_f}}{\sqrt {a \left (-b^{2}+\left (\textit {\_f}^{2}+x^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, x +\left (-a +b \right ) \textit {\_f}^{2}+b \left (x^{2}+a -b \right )}d \textit {\_f} +c_{1} &= 0 \\ -\left (\int _{\textit {\_b}}^{x}\frac {b \textit {\_a} -\sqrt {a \left (\left (-a +b \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )\right )}}{-\sqrt {a \left (\left (-a +b \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )\right )}\, \textit {\_a} +\left (-a +b \right ) y \left (x \right )^{2}+b \left (\textit {\_a}^{2}+a -b \right )}d \textit {\_a} \right )+\int _{}^{y \left (x \right )}\frac {\left (\left (-\sqrt {a \left (-b^{2}+\left (\textit {\_f}^{2}+x^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, x +\left (-a +b \right ) \textit {\_f}^{2}+b \left (x^{2}+a -b \right )\right ) \left (\int _{\textit {\_b}}^{x}-\frac {\left (a -b \right ) \left (-2 b \textit {\_a} \sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}+a \left (-b^{2}+\left (2 \textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )\right )}{\sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, {\left (-\sqrt {a \left (-b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, \textit {\_a} -b^{2}+\left (\textit {\_a}^{2}+\textit {\_f}^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}^{2}}d \textit {\_a} \right )+a -b \right ) \textit {\_f}}{-\sqrt {a \left (-b^{2}+\left (\textit {\_f}^{2}+x^{2}+a \right ) b -a \,\textit {\_f}^{2}\right )}\, x +\left (-a +b \right ) \textit {\_f}^{2}+b \left (x^{2}+a -b \right )}d \textit {\_f} +c_{1} &= 0 \\ \end{align*} |
Problem 4231
| ODE |
\[ \boxed {a^{2} \left (b^{2}-\left (x c -a y\right )^{2}\right ) {y^{\prime }}^{2}+2 a \,b^{2} c y^{\prime }+c^{2} \left (b^{2}-\left (x c -a y\right )^{2}\right )=0} \] |
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program solution |
\[ y = \frac {c^{3} x +\sqrt {b^{2} c^{4}}\, \sqrt {2}}{c^{2} a} \] Verified OK. \[ x = -\frac {c \sqrt {2}\, \left (\sqrt {-\left (a y-x c \right )^{2} \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-2 b^{2}\right )}+\left (x c -a y\right )^{2}\right ) b^{2} \left (b^{2}+\sqrt {-\left (a y-x c \right )^{2} \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-2 b^{2}\right )}\right )}{2 \sqrt {\frac {b^{4} c^{4} \left (b^{2}+\sqrt {-\left (a y-x c \right )^{2} \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-2 b^{2}\right )}\right ) \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}+\sqrt {-\left (a y-x c \right )^{2} \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-2 b^{2}\right )}\right )^{2}}{\left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-b^{2}\right )^{4}}}\, \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-b^{2}\right )^{2}}+c_{2} \] Verified OK. \[ x = \frac {c \sqrt {2}\, b^{2} \left (-b^{2}+\sqrt {-\left (a y-x c \right )^{2} \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-2 b^{2}\right )}\right ) \left (-\sqrt {-\left (a y-x c \right )^{2} \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-2 b^{2}\right )}+\left (x c -a y\right )^{2}\right )}{2 \sqrt {-\frac {b^{4} c^{4} \left (-b^{2}+\sqrt {-\left (a y-x c \right )^{2} \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-2 b^{2}\right )}\right ) \left (-a^{2} y^{2}+2 y c x a -c^{2} x^{2}+\sqrt {-\left (a y-x c \right )^{2} \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-2 b^{2}\right )}\right )^{2}}{\left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-b^{2}\right )^{4}}}\, \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-b^{2}\right )^{2}}+c_{2} \] Verified OK. \[ y = \frac {c^{3} x -\sqrt {b^{2} c^{4}}\, \sqrt {2}}{a \,c^{2}} \] Verified OK. \[ x = \frac {c \sqrt {2}\, \left (\sqrt {-\left (a y-x c \right )^{2} \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-2 b^{2}\right )}+\left (x c -a y\right )^{2}\right ) b^{2} \left (b^{2}+\sqrt {-\left (a y-x c \right )^{2} \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-2 b^{2}\right )}\right )}{2 \sqrt {\frac {b^{4} c^{4} \left (b^{2}+\sqrt {-\left (a y-x c \right )^{2} \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-2 b^{2}\right )}\right ) \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}+\sqrt {-\left (a y-x c \right )^{2} \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-2 b^{2}\right )}\right )^{2}}{\left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-b^{2}\right )^{4}}}\, \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-b^{2}\right )^{2}}+c_{4} \] Verified OK. \[ x = -\frac {c \sqrt {2}\, b^{2} \left (-b^{2}+\sqrt {-\left (a y-x c \right )^{2} \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-2 b^{2}\right )}\right ) \left (-\sqrt {-\left (a y-x c \right )^{2} \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-2 b^{2}\right )}+\left (x c -a y\right )^{2}\right )}{2 \sqrt {-\frac {b^{4} c^{4} \left (-b^{2}+\sqrt {-\left (a y-x c \right )^{2} \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-2 b^{2}\right )}\right ) \left (-a^{2} y^{2}+2 y c x a -c^{2} x^{2}+\sqrt {-\left (a y-x c \right )^{2} \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-2 b^{2}\right )}\right )^{2}}{\left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-b^{2}\right )^{4}}}\, \left (a^{2} y^{2}-2 y c x a +c^{2} x^{2}-b^{2}\right )^{2}}+c_{4} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {c x -\sqrt {2}\, b}{a} \\ y \left (x \right ) &= \frac {c x +\sqrt {2}\, b}{a} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (-a \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2} a^{2}-2 b^{2}+\sqrt {-a^{2} \textit {\_a}^{2} \left (\textit {\_a}^{2} a^{2}-2 b^{2}\right )}}{\textit {\_a}^{2} a^{2}-2 b^{2}}d \textit {\_a} \right )+2 c_{1} c -2 c x \right ) a +c x}{a} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (a \left (\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{2} a^{2}-2 b^{2}-\sqrt {-a^{2} \textit {\_a}^{2} \left (\textit {\_a}^{2} a^{2}-2 b^{2}\right )}}{\textit {\_a}^{2} a^{2}-2 b^{2}}d \textit {\_a} \right )+2 c_{1} c -2 c x \right ) a +c x}{a} \\ \end{align*} |
Problem 4232
| ODE |
\[ \boxed {x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }=-a^{2} x} \] |
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program solution |
\[ -\frac {\ln \left (y^{2}+\sqrt {y^{4}-4 a^{2} x^{2}}\right )}{2} = -\ln \left (x \right )+c_{1} \] Verified OK. \[ \frac {\ln \left (y^{2}+\sqrt {y^{4}-4 a^{2} x^{2}}\right )}{2} = c_{1} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \sqrt {2}\, \sqrt {-a x} \\ y \left (x \right ) &= -\sqrt {2}\, \sqrt {-a x} \\ y \left (x \right ) &= \sqrt {2}\, \sqrt {a x} \\ y \left (x \right ) &= -\sqrt {2}\, \sqrt {a x} \\ y \left (x \right ) &= \frac {{\mathrm e}^{\frac {c_{1}}{2}+\frac {\operatorname {RootOf}\left (16 x \,a^{2} {\mathrm e}^{2 \textit {\_Z} +2 c_{1}}+{\mathrm e}^{2 \textit {\_Z}} x^{3}-4 \,{\mathrm e}^{2 c_{1} +3 \textit {\_Z}}\right )}{2}}}{\sqrt {x}} \\ y \left (x \right ) &= \sqrt {x}\, {\mathrm e}^{-\frac {c_{1}}{2}+\frac {\operatorname {RootOf}\left (x^{2} \left (16 a^{2} x^{2} {\mathrm e}^{2 \textit {\_Z} -2 c_{1}}-4 \,{\mathrm e}^{3 \textit {\_Z} -2 c_{1}} x +{\mathrm e}^{2 \textit {\_Z}}\right )\right )}{2}} \\ \end{align*} |
Problem 4233
| ODE |
\[ \boxed {x y^{2} {y^{\prime }}^{2}+\left (a -x^{3}-y^{3}\right ) y^{\prime }+y x^{2}=0} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= \left (x^{3}+a -2 x \sqrt {a x}\right )^{\frac {1}{3}} \\ y \left (x \right ) &= \left (x^{3}+a +2 x \sqrt {a x}\right )^{\frac {1}{3}} \\ y \left (x \right ) &= -\frac {\left (x^{3}+a -2 x \sqrt {a x}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2} \\ y \left (x \right ) &= \frac {\left (x^{3}+a -2 x \sqrt {a x}\right )^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )}{2} \\ y \left (x \right ) &= -\frac {\left (x^{3}+a +2 x \sqrt {a x}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2} \\ y \left (x \right ) &= \frac {\left (x^{3}+a +2 x \sqrt {a x}\right )^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )}{2} \\ y \left (x \right ) &= 0 \\ \int _{\textit {\_b}}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{6}+\left (-2 x^{3}-2 a \right ) \textit {\_a}^{3}+\left (-x^{3}+a \right )^{2}}}d \textit {\_a} +\frac {\ln \left (x \right )}{2}-c_{1} &= 0 \\ \int _{\textit {\_b}}^{y \left (x \right )}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{6}+\left (-2 x^{3}-2 a \right ) \textit {\_a}^{3}+\left (-x^{3}+a \right )^{2}}}d \textit {\_a} -\frac {\ln \left (x \right )}{2}-c_{1} &= 0 \\ \end{align*} |
Problem 4234
| ODE |
\[ \boxed {2 x y^{2} {y^{\prime }}^{2}-y^{3} y^{\prime }=a} \] |
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program solution |
\[ -\frac {\ln \left (y^{2}+\sqrt {y^{4}+8 a x}\right )}{2} = -\frac {\ln \left (x \right )}{2}+c_{1} \] Verified OK. \[ \frac {\ln \left (y^{2}+\sqrt {y^{4}+8 a x}\right )}{2} = c_{1} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= 2^{\frac {3}{4}} \left (-a x \right )^{\frac {1}{4}} \\ y \left (x \right ) &= -2^{\frac {3}{4}} \left (-a x \right )^{\frac {1}{4}} \\ y \left (x \right ) &= -i 2^{\frac {3}{4}} \left (-a x \right )^{\frac {1}{4}} \\ y \left (x \right ) &= i 2^{\frac {3}{4}} \left (-a x \right )^{\frac {1}{4}} \\ y \left (x \right ) &= \frac {2^{\frac {1}{4}} \left (a \left (c_{1} -x \right )^{2} c_{1}^{3}\right )^{\frac {1}{4}}}{c_{1}} \\ y \left (x \right ) &= -\frac {2^{\frac {1}{4}} \left (a \left (c_{1} -x \right )^{2} c_{1}^{3}\right )^{\frac {1}{4}}}{c_{1}} \\ y \left (x \right ) &= -\frac {i 2^{\frac {1}{4}} \left (a \left (c_{1} -x \right )^{2} c_{1}^{3}\right )^{\frac {1}{4}}}{c_{1}} \\ y \left (x \right ) &= \frac {i 2^{\frac {1}{4}} \left (a \left (c_{1} -x \right )^{2} c_{1}^{3}\right )^{\frac {1}{4}}}{c_{1}} \\ \end{align*} |
Problem 4235
| ODE |
\[ \boxed {4 x^{2} y^{2} {y^{\prime }}^{2}-\left (x^{2}+y^{2}\right )^{2}=0} \] |
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program solution |
\[ \frac {3^{\frac {5}{6}} \left (\frac {3 y^{2}+x^{2}}{x^{2}}\right )^{\frac {1}{6}}}{3} = \frac {c_{6} {\mathrm e}^{c_{5}}}{\sqrt {x}} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \sqrt {\left (x +c_{1} \right ) x} \\ y \left (x \right ) &= -\sqrt {\left (x +c_{1} \right ) x} \\ y \left (x \right ) &= -\frac {\sqrt {3}\, \sqrt {-x \left (x^{3}-3 c_{1} \right )}}{3 x} \\ y \left (x \right ) &= \frac {\sqrt {3}\, \sqrt {-x \left (x^{3}-3 c_{1} \right )}}{3 x} \\ \end{align*} |
Problem 4236
| ODE |
\[ \boxed {4 y^{3} {y^{\prime }}^{2}-4 x y^{\prime }+y=0} \] |
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program solution |
\[ \frac {\ln \left (2\right )}{2}+\frac {\ln \left (x \right )}{2}+\frac {\ln \left (x +\sqrt {x^{2}-y^{4}}\right )}{2} = \frac {\ln \left (x \right )}{2}+c_{1} \] Verified OK. \[ y = {\mathrm e}^{\frac {\ln \left (2\right )}{4}+\frac {\ln \left (-2 \,{\mathrm e}^{4 c_{1}} {\mathrm e}^{-2 c_{1}}+2 \,{\mathrm e}^{-2 c_{1}} {\mathrm e}^{2 c_{1}} x \right )}{4}+\frac {c_{1}}{2}} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \sqrt {-x} \\ y \left (x \right ) &= -\sqrt {-x} \\ y \left (x \right ) &= \sqrt {x} \\ y \left (x \right ) &= -\sqrt {x} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )-2 \left (\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{4}+\sqrt {-\textit {\_a}^{4}+1}-1}{\textit {\_a} \left (\textit {\_a}^{4}-1\right )}d \textit {\_a} \right )+c_{1} \right ) \sqrt {x} \\ \end{align*} |
Problem 4237
| ODE |
\[ \boxed {3 x y^{4} {y^{\prime }}^{2}-y^{5} y^{\prime }=-1} \] |
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program solution |
\[ \frac {\ln \left (x \right )}{6} = \int _{}^{\frac {y}{x^{\frac {1}{6}}}}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{6}-12}}d \textit {\_a} +c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= 2^{\frac {1}{3}} 3^{\frac {1}{6}} x^{\frac {1}{6}} \\ y \left (x \right ) &= -2^{\frac {1}{3}} 3^{\frac {1}{6}} x^{\frac {1}{6}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) 3^{\frac {1}{6}} 2^{\frac {1}{3}} x^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {\left (-1+i \sqrt {3}\right ) 3^{\frac {1}{6}} 2^{\frac {1}{3}} x^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= -\frac {\left (-1+i \sqrt {3}\right ) 3^{\frac {1}{6}} 2^{\frac {1}{3}} x^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {\left (1+i \sqrt {3}\right ) 3^{\frac {1}{6}} 2^{\frac {1}{3}} x^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {3^{\frac {1}{6}} \left (-\left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{c_{1}} \\ y \left (x \right ) &= -\frac {3^{\frac {1}{6}} \left (-\left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{c_{1}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) 3^{\frac {1}{6}} \left (-\left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{2 c_{1}} \\ y \left (x \right ) &= \frac {\left (i 3^{\frac {2}{3}}-3^{\frac {1}{6}}\right ) \left (-\left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{2 c_{1}} \\ y \left (x \right ) &= -\frac {\left (-1+i \sqrt {3}\right ) 3^{\frac {1}{6}} \left (-\left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{2 c_{1}} \\ y \left (x \right ) &= \frac {\left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) \left (-\left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{2 c_{1}} \\ \end{align*} |
Problem 4238
| ODE |
\[ \boxed {9 x y^{4} {y^{\prime }}^{2}-3 y^{5} y^{\prime }=a} \] |
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program solution |
\[ \frac {\ln \left (x \right )}{6} = \int _{}^{\frac {y}{x^{\frac {1}{6}}}}\frac {\textit {\_a}^{2}}{\sqrt {\textit {\_a}^{6}+4 a}}d \textit {\_a} +c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= 2^{\frac {1}{3}} \left (-a x \right )^{\frac {1}{6}} \\ y \left (x \right ) &= -2^{\frac {1}{3}} \left (-a x \right )^{\frac {1}{6}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-a x \right )^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {\left (-1+i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-a x \right )^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= -\frac {\left (-1+i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-a x \right )^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {\left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-a x \right )^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {\left (a \left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{c_{1}} \\ y \left (x \right ) &= -\frac {\left (a \left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{c_{1}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (a \left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{2 c_{1}} \\ y \left (x \right ) &= \frac {\left (-1+i \sqrt {3}\right ) \left (a \left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{2 c_{1}} \\ y \left (x \right ) &= -\frac {\left (-1+i \sqrt {3}\right ) \left (a \left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{2 c_{1}} \\ y \left (x \right ) &= \frac {\left (1+i \sqrt {3}\right ) \left (a \left (c_{1} -x \right )^{2} c_{1}^{5}\right )^{\frac {1}{6}}}{2 c_{1}} \\ \end{align*} |
Problem 4239
| ODE |
\[ \boxed {9 \left (-x^{2}+1\right ) y^{4} {y^{\prime }}^{2}+6 x y^{5} y^{\prime }=-4 x^{2}} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= 2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}} \\ y \left (x \right ) &= -2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {\left (-1+i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= -\frac {\left (-1+i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {\left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {1}{6}}}{2} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} {\left (\left (-4 c_{1}^{2}+x^{2}-1\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{2 c_{1}} \\ y \left (x \right ) &= -\frac {2^{\frac {2}{3}} {\left (\left (-4 c_{1}^{2}+x^{2}-1\right ) c_{1}^{2}\right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{4 c_{1}} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} {\left (\left (-4 c_{1}^{2}+x^{2}-1\right ) c_{1}^{2}\right )}^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )}{4 c_{1}} \\ \end{align*} |
Problem 4240
| ODE |
\[ \boxed {{y^{\prime }}^{3}=b x +a} \] |
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program solution |
\[ y = \frac {3 \left (b x +a \right )^{\frac {4}{3}}}{4 b}+c_{1} \] Verified OK. \[ y = \frac {3 \left (b x +a \right )^{\frac {4}{3}} \left (i \sqrt {3}-1\right )}{8 b}+c_{2} \] Verified OK. \[ y = -\frac {3 \left (b x +a \right )^{\frac {4}{3}} \left (1+i \sqrt {3}\right )}{8 b}+c_{3} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {\left (3 b x +3 a \right ) \left (b x +a \right )^{\frac {1}{3}}+4 c_{1} b}{4 b} \\ y \left (x \right ) &= \frac {-3 \left (b x +a \right )^{\frac {4}{3}} \left (1+i \sqrt {3}\right )+8 c_{1} b}{8 b} \\ y \left (x \right ) &= \frac {3 \left (b x +a \right )^{\frac {4}{3}} \left (-1+i \sqrt {3}\right )+8 c_{1} b}{8 b} \\ \end{align*} |
Problem 4241
| ODE |
\[ \boxed {{y^{\prime }}^{3}=a \,x^{n}} \] |
|
program solution |
\[ y = \frac {3 x \left (a \,x^{n}\right )^{\frac {1}{3}}}{n +3}+c_{1} \] Verified OK. \[ y = \frac {3 x \left (a \,x^{n}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{2 \left (n +3\right )}+c_{2} \] Verified OK. \[ y = -\frac {3 x \left (a \,x^{n}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2 \left (n +3\right )}+c_{3} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {3 x \left (a \,x^{n}\right )^{\frac {1}{3}}+c_{1} \left (n +3\right )}{n +3} \\ y \left (x \right ) &= \frac {\left (3 i \sqrt {3}\, x -3 x \right ) \left (a \,x^{n}\right )^{\frac {1}{3}}+2 c_{1} \left (n +3\right )}{2 n +6} \\ y \left (x \right ) &= \frac {\left (-3 i \sqrt {3}\, x -3 x \right ) \left (a \,x^{n}\right )^{\frac {1}{3}}+2 c_{1} \left (n +3\right )}{2 n +6} \\ \end{align*} |
Problem 4242
| ODE |
\[ \boxed {{y^{\prime }}^{3}-y=-x} \] |
|
program solution |
\[ y = x +1 \] Verified OK. \[ x = \frac {3 \left (y-x \right )^{\frac {2}{3}}}{2}+3 \left (y-x \right )^{\frac {1}{3}}+3 \ln \left (\left (y-x \right )^{\frac {1}{3}}-1\right )+c_{2} \] Verified OK. \[ x = -\frac {3 \left (y-x \right )^{\frac {2}{3}}}{4}+\frac {3 i \sqrt {3}\, \left (y-x \right )^{\frac {2}{3}}}{4}-\frac {3 \left (y-x \right )^{\frac {1}{3}}}{2}-\frac {3 i \sqrt {3}\, \left (y-x \right )^{\frac {1}{3}}}{2}-3 \ln \left (2\right )+3 \ln \left (-\left (y-x \right )^{\frac {1}{3}}-i \sqrt {3}\, \left (y-x \right )^{\frac {1}{3}}-2\right )+c_{2} \] Verified OK. \[ x = -\frac {3 \left (y-x \right )^{\frac {2}{3}}}{4}-\frac {3 i \sqrt {3}\, \left (y-x \right )^{\frac {2}{3}}}{4}-\frac {3 \left (y-x \right )^{\frac {1}{3}}}{2}+\frac {3 i \sqrt {3}\, \left (y-x \right )^{\frac {1}{3}}}{2}-3 \ln \left (2\right )+3 \ln \left (i \sqrt {3}\, \left (y-x \right )^{\frac {1}{3}}-\left (y-x \right )^{\frac {1}{3}}-2\right )+c_{2} \] Verified OK.
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Maple solution |
\begin{align*} x -\frac {3 \left (y \left (x \right )-x \right )^{\frac {2}{3}}}{2}-3 \left (y \left (x \right )-x \right )^{\frac {1}{3}}-3 \ln \left (\left (y \left (x \right )-x \right )^{\frac {1}{3}}-1\right )-c_{1} &= 0 \\ x +\frac {3 \left (y \left (x \right )-x \right )^{\frac {2}{3}}}{4}-\frac {3 i \sqrt {3}\, \left (y \left (x \right )-x \right )^{\frac {2}{3}}}{4}+\frac {3 \left (y \left (x \right )-x \right )^{\frac {1}{3}}}{2}+\frac {3 i \sqrt {3}\, \left (y \left (x \right )-x \right )^{\frac {1}{3}}}{2}+6 \ln \left (2\right )-3 \ln \left (-4-2 i \sqrt {3}\, \left (y \left (x \right )-x \right )^{\frac {1}{3}}-2 \left (y \left (x \right )-x \right )^{\frac {1}{3}}\right )-c_{1} &= 0 \\ x +\frac {3 \left (y \left (x \right )-x \right )^{\frac {2}{3}}}{4}+\frac {3 i \sqrt {3}\, \left (y \left (x \right )-x \right )^{\frac {2}{3}}}{4}+\frac {3 \left (y \left (x \right )-x \right )^{\frac {1}{3}}}{2}-\frac {3 i \sqrt {3}\, \left (y \left (x \right )-x \right )^{\frac {1}{3}}}{2}+6 \ln \left (2\right )-3 \ln \left (2 i \sqrt {3}\, \left (y \left (x \right )-x \right )^{\frac {1}{3}}-2 \left (y \left (x \right )-x \right )^{\frac {1}{3}}-4\right )-c_{1} &= 0 \\ \end{align*} |
Problem 4243
| ODE |
\[ \boxed {{y^{\prime }}^{3}-\left (a +b y+y^{2} c \right ) f \left (x \right )=0} \] |
|
program solution |
\[
\int _{}^{y}\frac {1}{\left (c \,y^{2}+b y +a \right )^{\frac {1}{3}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{3}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}\frac {2}{\left (c \,y^{2}+b y +a \right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{3}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {2}{\left (c \,y^{2}+b y +a \right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{3}}d x +c_{1}
\] Verified OK.
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|
Maple solution |
\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a}^{2} c +b \textit {\_a} +a \right )^{\frac {1}{3}}}d \textit {\_a} -\frac {\int _{}^{x}{\left (\left (a +b y \left (x \right )+c y \left (x \right )^{2}\right ) f \left (\textit {\_a} \right )\right )}^{\frac {1}{3}}d \textit {\_a}}{\left (a +b y \left (x \right )+c y \left (x \right )^{2}\right )^{\frac {1}{3}}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a}^{2} c +b \textit {\_a} +a \right )^{\frac {1}{3}}}d \textit {\_a} +\frac {\left (1+i \sqrt {3}\right ) \left (\int _{}^{x}{\left (\left (a +b y \left (x \right )+c y \left (x \right )^{2}\right ) f \left (\textit {\_a} \right )\right )}^{\frac {1}{3}}d \textit {\_a} \right )}{2 \left (a +b y \left (x \right )+c y \left (x \right )^{2}\right )^{\frac {1}{3}}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a}^{2} c +b \textit {\_a} +a \right )^{\frac {1}{3}}}d \textit {\_a} -\frac {\left (-1+i \sqrt {3}\right ) \left (\int _{}^{x}{\left (\left (a +b y \left (x \right )+c y \left (x \right )^{2}\right ) f \left (\textit {\_a} \right )\right )}^{\frac {1}{3}}d \textit {\_a} \right )}{2 \left (a +b y \left (x \right )+c y \left (x \right )^{2}\right )^{\frac {1}{3}}}+c_{1} &= 0 \\ \end{align*} |
Problem 4244
| ODE |
\[ \boxed {{y^{\prime }}^{3}-\left (y-a \right )^{2} \left (y-b \right )^{2}=0} \] |
|
program solution |
\[ \int _{}^{y}\frac {1}{\left (\textit {\_a}^{4}-2 \textit {\_a}^{3} a -2 \textit {\_a}^{3} b +\textit {\_a}^{2} a^{2}+4 \textit {\_a}^{2} a b +\textit {\_a}^{2} b^{2}-2 \textit {\_a} \,a^{2} b -2 \textit {\_a} a \,b^{2}+a^{2} b^{2}\right )^{\frac {1}{3}}}d \textit {\_a} = x +c_{1} \] Verified OK. \[ \int _{}^{y}\frac {1}{-\frac {\left (\textit {\_a}^{4}-2 \textit {\_a}^{3} a -2 \textit {\_a}^{3} b +\textit {\_a}^{2} a^{2}+4 \textit {\_a}^{2} a b +\textit {\_a}^{2} b^{2}-2 \textit {\_a} \,a^{2} b -2 \textit {\_a} a \,b^{2}+a^{2} b^{2}\right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (\textit {\_a}^{4}-2 \textit {\_a}^{3} a -2 \textit {\_a}^{3} b +\textit {\_a}^{2} a^{2}+4 \textit {\_a}^{2} a b +\textit {\_a}^{2} b^{2}-2 \textit {\_a} \,a^{2} b -2 \textit {\_a} a \,b^{2}+a^{2} b^{2}\right )^{\frac {1}{3}}}{2}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}\frac {1}{-\frac {\left (\textit {\_a}^{4}-2 \textit {\_a}^{3} a -2 \textit {\_a}^{3} b +\textit {\_a}^{2} a^{2}+4 \textit {\_a}^{2} a b +\textit {\_a}^{2} b^{2}-2 \textit {\_a} \,a^{2} b -2 \textit {\_a} a \,b^{2}+a^{2} b^{2}\right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (\textit {\_a}^{4}-2 \textit {\_a}^{3} a -2 \textit {\_a}^{3} b +\textit {\_a}^{2} a^{2}+4 \textit {\_a}^{2} a b +\textit {\_a}^{2} b^{2}-2 \textit {\_a} \,a^{2} b -2 \textit {\_a} a \,b^{2}+a^{2} b^{2}\right )^{\frac {1}{3}}}{2}}d \textit {\_a} = x +c_{3} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= a \\ y \left (x \right ) &= b \\ x -\left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{2} \left (\textit {\_a} -b \right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} \right )-c_{1} &= 0 \\ \frac {2 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{2} \left (\textit {\_a} -b \right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+x -c_{1}}{1+i \sqrt {3}} &= 0 \\ \frac {-2 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{2} \left (\textit {\_a} -b \right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}-x +c_{1}}{-1+i \sqrt {3}} &= 0 \\ \end{align*} |
Problem 4245
| ODE |
\[ \boxed {{y^{\prime }}^{3}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right )^{2}=0} \] |
|
program solution |
\[
\int _{}^{y}\frac {1}{\left (\left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}\frac {2}{\left (\left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {2}{\left (\left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1}
\] Verified OK.
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Maple solution |
\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )\right )^{\frac {2}{3}}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{2} \left (y \left (x \right )-b \right )^{2}\right )^{\frac {1}{3}}d \textit {\_a}}{\left (\left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {2}{3}}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )\right )^{\frac {2}{3}}}d \textit {\_a} +\frac {\left (1+i \sqrt {3}\right ) \left (\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{2} \left (y \left (x \right )-b \right )^{2}\right )^{\frac {1}{3}}d \textit {\_a} \right )}{2 \left (\left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {2}{3}}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )\right )^{\frac {2}{3}}}d \textit {\_a} -\frac {\left (-1+i \sqrt {3}\right ) \left (\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{2} \left (y \left (x \right )-b \right )^{2}\right )^{\frac {1}{3}}d \textit {\_a} \right )}{2 \left (\left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {2}{3}}}+c_{1} &= 0 \\ \end{align*} |
Problem 4246
| ODE |
\[ \boxed {{y^{\prime }}^{3}+f \left (x \right ) \left (y-a \right )^{2} \left (y-b \right )^{2} \left (y-c \right )^{2}=0} \] |
|
program solution |
\[
\int _{}^{y}\frac {1}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}\frac {2}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {2}{\left (\left (c -y \right )^{2} \left (b -y \right )^{2} \left (a -y \right )^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{3}}d x +c_{1}
\] Verified OK.
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Maple solution |
\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -c \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )\right )^{\frac {2}{3}}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-c \right )^{2} \left (y \left (x \right )-b \right )^{2} \left (y \left (x \right )-a \right )^{2}\right )^{\frac {1}{3}}d \textit {\_a}}{\left (\left (y \left (x \right )-c \right ) \left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {2}{3}}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -c \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )\right )^{\frac {2}{3}}}d \textit {\_a} +\frac {\left (1+i \sqrt {3}\right ) \left (\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-c \right )^{2} \left (y \left (x \right )-b \right )^{2} \left (y \left (x \right )-a \right )^{2}\right )^{\frac {1}{3}}d \textit {\_a} \right )}{2 \left (\left (y \left (x \right )-c \right ) \left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {2}{3}}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -c \right ) \left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )\right )^{\frac {2}{3}}}d \textit {\_a} -\frac {\left (-1+i \sqrt {3}\right ) \left (\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-c \right )^{2} \left (y \left (x \right )-b \right )^{2} \left (y \left (x \right )-a \right )^{2}\right )^{\frac {1}{3}}d \textit {\_a} \right )}{2 \left (\left (y \left (x \right )-c \right ) \left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {2}{3}}}+c_{1} &= 0 \\ \end{align*} |
Problem 4247
| ODE |
\[ \boxed {{y^{\prime }}^{3}+y^{\prime }=b x -a} \] |
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program solution |
\[ y = \int \frac {\left (108 b x -108 a +12 \sqrt {81 x^{2} b^{2}-162 a b x +81 a^{2}+12}\right )^{\frac {2}{3}}-12}{6 \left (108 b x -108 a +12 \sqrt {81 x^{2} b^{2}-162 a b x +81 a^{2}+12}\right )^{\frac {1}{3}}}d x +c_{1} \] Verified OK. \[ y = \int \frac {i \left (108 b x -108 a +12 \sqrt {81 x^{2} b^{2}-162 a b x +81 a^{2}+12}\right )^{\frac {2}{3}} \sqrt {3}+12 i \sqrt {3}-\left (108 b x -108 a +12 \sqrt {81 x^{2} b^{2}-162 a b x +81 a^{2}+12}\right )^{\frac {2}{3}}+12}{12 \left (108 b x -108 a +12 \sqrt {81 x^{2} b^{2}-162 a b x +81 a^{2}+12}\right )^{\frac {1}{3}}}d x +c_{2} \] Verified OK. \[ y = \int -\frac {i \left (108 b x -108 a +12 \sqrt {81 x^{2} b^{2}-162 a b x +81 a^{2}+12}\right )^{\frac {2}{3}} \sqrt {3}+12 i \sqrt {3}+\left (108 b x -108 a +12 \sqrt {81 x^{2} b^{2}-162 a b x +81 a^{2}+12}\right )^{\frac {2}{3}}-12}{12 \left (108 b x -108 a +12 \sqrt {81 x^{2} b^{2}-162 a b x +81 a^{2}+12}\right )^{\frac {1}{3}}}d x +c_{3} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -\frac {\left (\int \frac {i \left (108 b x -108 a +12 \sqrt {81 b^{2} x^{2}-162 a b x +81 a^{2}+12}\right )^{\frac {2}{3}} \sqrt {3}+\left (108 b x -108 a +12 \sqrt {81 b^{2} x^{2}-162 a b x +81 a^{2}+12}\right )^{\frac {2}{3}}+12 i \sqrt {3}-12}{\left (108 b x -108 a +12 \sqrt {81 b^{2} x^{2}-162 a b x +81 a^{2}+12}\right )^{\frac {1}{3}}}d x \right )}{12}+c_{1} \\ y \left (x \right ) &= \frac {\left (\int \frac {\left (-1+i \sqrt {3}\right ) \left (108 b x -108 a +12 \sqrt {81 b^{2} x^{2}-162 a b x +81 a^{2}+12}\right )^{\frac {2}{3}}+12 i \sqrt {3}+12}{\left (108 b x -108 a +12 \sqrt {81 b^{2} x^{2}-162 a b x +81 a^{2}+12}\right )^{\frac {1}{3}}}d x \right )}{12}+c_{1} \\ y \left (x \right ) &= \frac {\left (\int \frac {\left (108 b x -108 a +12 \sqrt {81 b^{2} x^{2}-162 a b x +81 a^{2}+12}\right )^{\frac {2}{3}}-12}{\left (108 b x -108 a +12 \sqrt {81 b^{2} x^{2}-162 a b x +81 a^{2}+12}\right )^{\frac {1}{3}}}d x \right )}{6}+c_{1} \\ \end{align*} |
Problem 4248
| ODE |
\[ \boxed {{y^{\prime }}^{3}+y^{\prime }-y=0} \] |
|
program solution |
\[ \int _{}^{y}\frac {6 \left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {1}{3}}}{\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {2}{3}}-12}d \textit {\_a} = x +c_{1} \] Verified OK. \[ \int _{}^{y}\frac {12 \left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {2}{3}}+12 i \sqrt {3}-\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {2}{3}}+12}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {12 \left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {2}{3}}-12}d \textit {\_a} = x +c_{3} \] Verified OK.
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Maple solution |
\begin{align*} x -6 \left (\int _{}^{y \left (x \right )}\frac {\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {1}{3}}}{\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {2}{3}}-12}d \textit {\_a} \right )-c_{1} &= 0 \\ \frac {-12 \left (\int _{}^{y \left (x \right )}\frac {\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {1}{3}}}{-6-6 i \sqrt {3}-\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {2}{3}}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+x -c_{1}}{1+i \sqrt {3}} &= 0 \\ \frac {12 \left (\int _{}^{y \left (x \right )}\frac {\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {1}{3}}}{-\left (108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}+12}\right )^{\frac {2}{3}}+\left (\sqrt {3}+3 i\right )^{2}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+c_{1} -x}{-1+i \sqrt {3}} &= 0 \\ \end{align*} |
Problem 4249
| ODE |
\[ \boxed {{y^{\prime }}^{3}+y^{\prime }-{\mathrm e}^{y}=0} \] |
|
program solution |
\[ \int _{}^{y}\frac {6 \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}-12}d \textit {\_a} = x +c_{1} \] Verified OK. \[ \int _{}^{y}\frac {12 \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}+12 i \sqrt {3}-\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}+12}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {12 \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}-12}d \textit {\_a} = x +c_{3} \] Verified OK.
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Maple solution |
\begin{align*} x -6 \left (\int _{}^{y \left (x \right )}\frac {\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}-12}d \textit {\_a} \right )-c_{1} &= 0 \\ \frac {-12 \left (\int _{}^{y \left (x \right )}\frac {\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{-\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}-6-6 i \sqrt {3}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+x -c_{1}}{1+i \sqrt {3}} &= 0 \\ \frac {12 \left (\int _{}^{y \left (x \right )}\frac {\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{-\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}+\left (\sqrt {3}+3 i\right )^{2}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+c_{1} -x}{-1+i \sqrt {3}} &= 0 \\ \end{align*} |
Problem 4250
| ODE |
\[ \boxed {{y^{\prime }}^{3}-7 y^{\prime }=-6} \] |
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program solution |
\[ y = -3 x +c_{3} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= 2 x +c_{1} \\ y \left (x \right ) &= x +c_{1} \\ y \left (x \right ) &= -3 x +c_{1} \\ \end{align*} |
Problem 4251
| ODE |
\[ \boxed {{y^{\prime }}^{3}-x y^{\prime }+a y=0} \] |
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program solution |
\[ y = 0 \] Verified OK. \[ x = -\frac {{\left (\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}+12 x \right )}^{2}}{\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}} \left (24 a -36\right )}+c_{1} 6^{-\frac {1}{a -1}} {\left (\frac {\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}+12 x}{\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {1}{a -1}} \] Warning, solution could not be verified \[ x = \frac {{\left (\left (\sqrt {3}+i\right ) \left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}+12 \left (i-\sqrt {3}\right ) x \right )}^{2}}{\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}} \left (96 a -144\right )}+c_{1} 12^{-\frac {1}{a -1}} {\left (\frac {i \sqrt {3}\, \left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}-12 i \sqrt {3}\, x -\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}-12 x}{\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {1}{a -1}} \] Warning, solution could not be verified \[ x = -\frac {{\left (i \sqrt {3}\, \left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}-12 i \sqrt {3}\, x +\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}+12 x \right )}^{2}}{\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}} \left (96 a -144\right )}+c_{1} {\left (\frac {-i \sqrt {3}\, \left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}+12 i \sqrt {3}\, x -\left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {2}{3}}-12 x}{12 \left (-108 a y+12 \sqrt {-12 x^{3}+81 a^{2} y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {1}{a -1}} \] Warning, solution could not be verified |
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Maple solution |
\begin{align*} -\frac {48 \left (6^{-\frac {1}{a -1}} c_{1} \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}} \left (a -\frac {3}{2}\right )^{2} {\left (\frac {\left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}}+12 x}{\left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {1}{3}}}\right )}^{\frac {1}{a -1}}-\frac {x \left (a -\frac {1}{2}\right ) \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}}}{24}+\left (\frac {3 y \left (x \right ) a}{16}-\frac {\sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}}{48}\right ) \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {1}{3}}-\frac {x^{2}}{4}\right )}{\left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}} \left (2 a -3\right )} &= 0 \\ \frac {192 c_{1} \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}} \left (a -\frac {3}{2}\right )^{2} 12^{-\frac {1}{a -1}} {\left (\frac {i \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}} \sqrt {3}-12 i \sqrt {3}\, x -\left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}}-12 x}{\left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {1}{3}}}\right )}^{\frac {1}{a -1}}+4 x \left (a -\frac {1}{2}\right ) \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}}+9 \left (1+i \sqrt {3}\right ) \left (y \left (x \right ) a -\frac {\sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}}{9}\right ) \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {1}{3}}+12 \left (-1+i \sqrt {3}\right ) x^{2}}{\left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}} \left (4 a -6\right )} &= 0 \\ -\frac {9 \left (\frac {64 c_{1} \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}} \left (a -\frac {3}{2}\right )^{2} {\left (-\frac {i \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}} \sqrt {3}-12 i \sqrt {3}\, x +\left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}}+12 x}{12 \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {1}{3}}}\right )}^{\frac {1}{a -1}}}{3}-\frac {4 x \left (a -\frac {1}{2}\right ) \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}}}{9}+\left (y \left (x \right ) a -\frac {\sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}}{9}\right ) \left (-1+i \sqrt {3}\right ) \left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {1}{3}}+\frac {4 \left (1+i \sqrt {3}\right ) x^{2}}{3}\right )}{\left (-108 y \left (x \right ) a +12 \sqrt {81 y \left (x \right )^{2} a^{2}-12 x^{3}}\right )^{\frac {2}{3}} \left (4 a -6\right )} &= 0 \\ \end{align*} |
Problem 4252
| ODE |
\[ \boxed {{y^{\prime }}^{3}+2 x y^{\prime }-y=0} \] |
|
program solution |
\[ y = 0 \] Verified OK. \[ x = -\frac {{\left (\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x \right )}^{2}}{48 \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {36 c_{1} \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{{\left (\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x \right )}^{2}} \] Verified OK. \[ x = \frac {3 {\left (\frac {\left (\sqrt {3}+i\right ) \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{24}+x \left (-i+\sqrt {3}\right )\right )}^{2}}{\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {144 c_{1} \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{{\left (i \sqrt {3}\, \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 i \sqrt {3}\, x -\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}} \] Verified OK. \[ x = \frac {3 {\left (\frac {\left (-i+\sqrt {3}\right ) \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{24}+x \left (\sqrt {3}+i\right )\right )}^{2}}{\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {144 c_{1} \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}{{\left (i \sqrt {3}\, \left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 i \sqrt {3}\, x +\left (108 y+12 \sqrt {96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x \right )}^{2}} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {2 \left (-2 x +\sqrt {x^{2}+3 c_{1}}\right ) \sqrt {-6 \sqrt {x^{2}+3 c_{1}}-6 x}}{9} \\ y \left (x \right ) &= -\frac {2 \left (-2 x +\sqrt {x^{2}+3 c_{1}}\right ) \sqrt {-6 \sqrt {x^{2}+3 c_{1}}-6 x}}{9} \\ y \left (x \right ) &= -\frac {2 \left (2 x +\sqrt {x^{2}+3 c_{1}}\right ) \sqrt {6 \sqrt {x^{2}+3 c_{1}}-6 x}}{9} \\ y \left (x \right ) &= \frac {2 \left (2 x +\sqrt {x^{2}+3 c_{1}}\right ) \sqrt {6 \sqrt {x^{2}+3 c_{1}}-6 x}}{9} \\ \end{align*} |
Problem 4253
| ODE |
\[ \boxed {{y^{\prime }}^{3}-2 x y^{\prime }-y=0} \] |
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program solution |
\[ y = 0 \] Verified OK. \[ x = \frac {{\left (\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}}{96 \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {c_{1} 6^{\frac {2}{3}}}{{\left (\frac {\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x}{\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}} \] Verified OK. \[ x = \frac {{\left (-i \sqrt {3}\, \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 i \sqrt {3}\, x +\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}}{384 \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {c_{1} 12^{\frac {2}{3}}}{{\left (\frac {i \sqrt {3}\, \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 i \sqrt {3}\, x -\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x}{\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}} \] Verified OK. \[ x = \frac {{\left (i \sqrt {3}\, \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 i \sqrt {3}\, x +\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}}{384 \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}}+\frac {2 c_{1} 18^{\frac {1}{3}}}{{\left (\frac {-i \sqrt {3}\, \left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}+24 i \sqrt {3}\, x -\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {2}{3}}-24 x}{\left (108 y+12 \sqrt {-96 x^{3}+81 y^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}} \] Warning, solution could not be verified |
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Maple solution |
\begin{align*} -\frac {c_{1}}{{\left (\frac {\left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}+24 x}{\left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}}+x -\frac {{\left (\left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}+24 x \right )}^{2}}{96 \left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}} &= 0 \\ -\frac {c_{1}}{{\left (\frac {i \sqrt {3}\, \left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}-24 i \sqrt {3}\, x -\left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}-24 x}{\left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}}+x +\frac {3 {\left (-\frac {\left (\sqrt {3}+i\right ) \left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}}{24}+x \left (-i+\sqrt {3}\right )\right )}^{2}}{2 \left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}} &= 0 \\ -\frac {12^{\frac {2}{3}} c_{1}}{{\left (\frac {-i \sqrt {3}\, \left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}+24 i \sqrt {3}\, x -\left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}-24 x}{\left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {1}{3}}}\right )}^{\frac {2}{3}}}+x +\frac {3 {\left (\frac {\left (i-\sqrt {3}\right ) \left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}}{24}+\left (\sqrt {3}+i\right ) x \right )}^{2}}{2 \left (108 y \left (x \right )+12 \sqrt {-96 x^{3}+81 y \left (x \right )^{2}}\right )^{\frac {2}{3}}} &= 0 \\ \end{align*} |
Problem 4254
| ODE |
\[ \boxed {{y^{\prime }}^{3}-a x y^{\prime }=-x^{3}} \] |
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program solution |
\[ y = \int \frac {\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {2}{3}}+12 a x}{6 \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {1}{3}}}d x +c_{1} \] Verified OK. \[ y = \int \frac {i \sqrt {3}\, \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {2}{3}}-12 i \sqrt {3}\, a x -\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {2}{3}}-12 a x}{12 \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {1}{3}}}d x +c_{2} \] Verified OK. \[ y = \int -\frac {i \sqrt {3}\, \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {2}{3}}-12 i \sqrt {3}\, a x +\left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {2}{3}}+12 a x}{12 \left (-108 x^{3}+12 \sqrt {-12 a^{3} x^{3}+81 x^{6}}\right )^{\frac {1}{3}}}d x +c_{3} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {\left (\int \left (\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )-\frac {12 a \left (1+i \sqrt {3}\right ) x}{\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {1}{3}}}\right )d x \right )}{12}+c_{1} \\ y \left (x \right ) &= -\frac {\left (\int \left (\left (1+i \sqrt {3}\right ) \left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {1}{3}}-\frac {12 a \left (-1+i \sqrt {3}\right ) x}{\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {1}{3}}}\right )d x \right )}{12}+c_{1} \\ y \left (x \right ) &= \frac {\left (\int \frac {\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {2}{3}}+12 a x}{\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{\frac {1}{3}}}d x \right )}{6}+c_{1} \\ \end{align*} |
Problem 4255
| ODE |
\[ \boxed {{y^{\prime }}^{3}+a x y^{\prime }-a y=0} \] |
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program solution |
\[ y = c_{1} x +\frac {c_{1}^{3}}{a} \] Verified OK. \[ y = \frac {2 \sqrt {3}\, \sqrt {-a x}\, x}{9} \] Verified OK. \[ y = -\frac {2 \sqrt {3}\, \sqrt {-a x}\, x}{9} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -\frac {2 \sqrt {3}\, \sqrt {-a x}\, x}{9} \\ y \left (x \right ) &= \frac {2 \sqrt {3}\, \sqrt {-a x}\, x}{9} \\ y \left (x \right ) &= \frac {c_{1} \left (a x +c_{1}^{2}\right )}{a} \\ \end{align*} |
Problem 4256
| ODE |
\[ \boxed {{y^{\prime }}^{3}-\left (b x +a \right ) y^{\prime }+b y=0} \] |
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program solution |
\[ y = c_{1} x +\frac {c_{1} \left (-c_{1}^{2}+a \right )}{b} \] Verified OK. \[ y = \frac {2 \sqrt {3 b x +3 a}\, \left (b x +a \right )}{9 b} \] Verified OK. \[ y = -\frac {2 \sqrt {3 b x +3 a}\, \left (b x +a \right )}{9 b} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -\frac {2 \sqrt {3 b x +3 a}\, \left (b x +a \right )}{9 b} \\ y \left (x \right ) &= \frac {2 \sqrt {3 b x +3 a}\, \left (b x +a \right )}{9 b} \\ y \left (x \right ) &= \frac {c_{1} \left (b x -c_{1}^{2}+a \right )}{b} \\ \end{align*} |
Problem 4257
| ODE |
\[ \boxed {{y^{\prime }}^{3}-2 y^{\prime } y+y^{2}=0} \] |
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program solution |
\[ \int _{}^{y}\frac {6 \left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{\left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {2}{3}}+24 \textit {\_a}}d \textit {\_a} = x +c_{1} \] Verified OK. \[ \int _{}^{y}-\frac {12 \left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{-i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {2}{3}}+24 i \sqrt {3}\, \textit {\_a} +\left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {2}{3}}+24 \textit {\_a}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}\frac {12 \left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{-i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {2}{3}}+24 i \sqrt {3}\, \textit {\_a} -\left (-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-96 \textit {\_a}^{3}}\right )^{\frac {2}{3}}-24 \textit {\_a}}d \textit {\_a} = x +c_{3} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= 0 \\ -2^{\frac {2}{3}} \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {\left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{2^{\frac {1}{3}} \left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{\frac {2}{3}}+4 \textit {\_a}}d \textit {\_a} \right )+x -c_{1} &= 0 \\ \frac {2 \,2^{\frac {2}{3}} \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {\left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{2^{\frac {1}{3}} \left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{\frac {2}{3}}-2 i \textit {\_a} \sqrt {3}-2 \textit {\_a}}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (1+i \sqrt {3}\right )}{1+i \sqrt {3}} &= 0 \\ \frac {2 i 2^{\frac {2}{3}} \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {\left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{\frac {1}{3}}}{2^{\frac {1}{3}} \left (-3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}-32 \textit {\_a}^{3}}\right )^{\frac {2}{3}}+2 i \textit {\_a} \sqrt {3}-2 \textit {\_a}}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (\sqrt {3}+i\right )}{\sqrt {3}+i} &= 0 \\ \end{align*} |
Problem 4258
| ODE |
\[ \boxed {{y^{\prime }}^{3}-a x y y^{\prime }+2 y^{2} a=0} \] |
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program solution |
\[ -9 a^{\frac {1}{3}} \left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) \left (\int _{}^{\frac {y}{x^{3}}}\frac {\left (3 \sqrt {3}\, \sqrt {\textit {\_a}}-\sqrt {27 \textit {\_a} -a}\right )^{\frac {1}{3}}}{\sqrt {\textit {\_a}}\, \left (3 \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) a^{\frac {2}{3}} \left (3 \sqrt {3}\, \sqrt {\textit {\_a}}-\sqrt {27 \textit {\_a} -a}\right )^{\frac {2}{3}}-9 \sqrt {\textit {\_a}}\, \left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) a^{\frac {1}{3}} \left (3 \sqrt {3}\, \sqrt {\textit {\_a}}-\sqrt {27 \textit {\_a} -a}\right )^{\frac {1}{3}}+2 \,3^{\frac {2}{3}} a \right )}d \textit {\_a} \right )-c_{1} +3 \ln \left (x \right ) = 0 \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {a \,x^{3}}{27} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\left (a c_{1} x -1\right )^{2}}{4 c_{1}^{3} a^{2}} \\ \end{align*} |
Problem 4259
| ODE |
\[ \boxed {{y^{\prime }}^{3}-x y^{4} y^{\prime }-y^{5}=0} \] |
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program solution |
\[ \ln \left (x \right ) = \int _{}^{y x^{\frac {3}{2}}}\frac {2 \left (i \sqrt {12 \textit {\_a}^{2}-81}+9\right )^{\frac {1}{3}} 18^{\frac {2}{3}}}{\left (2 \textit {\_a}^{\frac {4}{3}} \left (i \sqrt {12 \textit {\_a}^{2}-81}+9\right )^{\frac {2}{3}} 18^{\frac {1}{3}}+9 \left (i \sqrt {12 \textit {\_a}^{2}-81}+9\right )^{\frac {1}{3}} 12^{\frac {1}{3}} \left (\textit {\_a}^{2}\right )^{\frac {1}{3}}+12 \textit {\_a}^{2}\right ) \textit {\_a}^{\frac {1}{3}}}d \textit {\_a} +c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {3 \sqrt {3}}{2 x^{\frac {3}{2}}} \\ y \left (x \right ) &= \frac {3 \sqrt {3}}{2 x^{\frac {3}{2}}} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= c_{1} \sqrt {\frac {c_{1}^{10}}{\left (c_{1}^{4} x -1\right )^{2}}} \\ \end{align*} |
Problem 4260
| ODE |
\[ \boxed {{y^{\prime }}^{3}+{\mathrm e}^{3 x -2 y} \left (y^{\prime }-1\right )=0} \] |
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program solution |
\[ y = \frac {3 x}{2}-\frac {3 \ln \left (3\right )}{2}+\ln \left (2\right )-\frac {i \pi }{2} \] Verified OK. \[ y = \frac {3 x}{2}-\frac {\ln \left (-\frac {{\mathrm e}^{3 x} c_{1}^{3}}{\left ({\mathrm e}^{x} c_{1} -1\right )^{3} \left (\frac {{\mathrm e}^{x} c_{1}}{{\mathrm e}^{x} c_{1} -1}-1\right )}\right )}{2} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {3 x}{2}+\operatorname {RootOf}\left (x +2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (\int _{}^{\textit {\_Z}}\frac {{\mathrm e}^{2 \textit {\_a}} {\left (\left (\sqrt {3}\, \sqrt {\left (4+27 \,{\mathrm e}^{2 \textit {\_a}}\right ) {\mathrm e}^{-6 \textit {\_a}}}\, {\mathrm e}^{2 \textit {\_a}}+9\right ) {\mathrm e}^{-2 \textit {\_a}}\right )}^{\frac {1}{3}}}{3 \,{\mathrm e}^{2 \textit {\_a}} 2^{\frac {1}{3}} 3^{\frac {2}{3}} {\left (\left (\sqrt {3}\, \sqrt {\left (4+27 \,{\mathrm e}^{2 \textit {\_a}}\right ) {\mathrm e}^{-6 \textit {\_a}}}\, {\mathrm e}^{2 \textit {\_a}}+9\right ) {\mathrm e}^{-2 \textit {\_a}}\right )}^{\frac {1}{3}}-2 {\left (\left (\sqrt {3}\, \sqrt {\left (4+27 \,{\mathrm e}^{2 \textit {\_a}}\right ) {\mathrm e}^{-6 \textit {\_a}}}\, {\mathrm e}^{2 \textit {\_a}}+9\right ) {\mathrm e}^{-2 \textit {\_a}}\right )}^{\frac {2}{3}} {\mathrm e}^{2 \textit {\_a}}+2 \,2^{\frac {2}{3}} 3^{\frac {1}{3}}}d \textit {\_a} \right )-c_{1} \right ) \\ y \left (x \right ) &= \frac {3 x}{2}+\operatorname {RootOf}\left (-2 \left (\int _{}^{\textit {\_Z}}\frac {{\mathrm e}^{2 \textit {\_a} +3} \left (2 \sqrt {{\mathrm e}^{-4 \textit {\_a}} \left (4 \,{\mathrm e}^{-2 \textit {\_a}}+27\right )}+6 \sqrt {3}\, {\mathrm e}^{-2 \textit {\_a}}\right )^{\frac {1}{3}}}{3 \,{\mathrm e}^{2 \textit {\_a} +3} \left (2 \sqrt {{\mathrm e}^{-4 \textit {\_a}} \left (4 \,{\mathrm e}^{-2 \textit {\_a}}+27\right )}+6 \sqrt {3}\, {\mathrm e}^{-2 \textit {\_a}}\right )^{\frac {1}{3}} 3^{\frac {5}{6}}+4 \,{\mathrm e}^{2 \textit {\_a} +3} 3^{\frac {1}{3}} {\left (\left (\sqrt {{\mathrm e}^{-4 \textit {\_a}} \left (4 \,{\mathrm e}^{-2 \textit {\_a}}+27\right )}+3 \sqrt {3}\, {\mathrm e}^{-2 \textit {\_a}}\right )^{2}\right )}^{\frac {1}{3}}-9 i {\mathrm e}^{2 \textit {\_a} +3} 3^{\frac {1}{3}} \left (2 \sqrt {{\mathrm e}^{-4 \textit {\_a}} \left (4 \,{\mathrm e}^{-2 \textit {\_a}}+27\right )}+6 \sqrt {3}\, {\mathrm e}^{-2 \textit {\_a}}\right )^{\frac {1}{3}}+2 \,2^{\frac {2}{3}} 3^{\frac {1}{3}} {\mathrm e}^{3}+2 i 2^{\frac {2}{3}} 3^{\frac {5}{6}} {\mathrm e}^{3}}d \textit {\_a} \right ) 3^{\frac {5}{6}}+6 i 3^{\frac {1}{3}} \left (\int _{}^{\textit {\_Z}}\frac {{\mathrm e}^{2 \textit {\_a} +3} \left (2 \sqrt {{\mathrm e}^{-4 \textit {\_a}} \left (4 \,{\mathrm e}^{-2 \textit {\_a}}+27\right )}+6 \sqrt {3}\, {\mathrm e}^{-2 \textit {\_a}}\right )^{\frac {1}{3}}}{3 \,{\mathrm e}^{2 \textit {\_a} +3} \left (2 \sqrt {{\mathrm e}^{-4 \textit {\_a}} \left (4 \,{\mathrm e}^{-2 \textit {\_a}}+27\right )}+6 \sqrt {3}\, {\mathrm e}^{-2 \textit {\_a}}\right )^{\frac {1}{3}} 3^{\frac {5}{6}}+4 \,{\mathrm e}^{2 \textit {\_a} +3} 3^{\frac {1}{3}} {\left (\left (\sqrt {{\mathrm e}^{-4 \textit {\_a}} \left (4 \,{\mathrm e}^{-2 \textit {\_a}}+27\right )}+3 \sqrt {3}\, {\mathrm e}^{-2 \textit {\_a}}\right )^{2}\right )}^{\frac {1}{3}}-9 i {\mathrm e}^{2 \textit {\_a} +3} 3^{\frac {1}{3}} \left (2 \sqrt {{\mathrm e}^{-4 \textit {\_a}} \left (4 \,{\mathrm e}^{-2 \textit {\_a}}+27\right )}+6 \sqrt {3}\, {\mathrm e}^{-2 \textit {\_a}}\right )^{\frac {1}{3}}+2 \,2^{\frac {2}{3}} 3^{\frac {1}{3}} {\mathrm e}^{3}+2 i 2^{\frac {2}{3}} 3^{\frac {5}{6}} {\mathrm e}^{3}}d \textit {\_a} \right )+c_{1} -x \right ) \\ y \left (x \right ) &= \frac {3 x}{2}+\operatorname {RootOf}\left (2 \left (\int _{}^{\textit {\_Z}}\frac {{\mathrm e}^{2 \textit {\_a} +3} \left (2 \sqrt {{\mathrm e}^{-4 \textit {\_a}} \left (4 \,{\mathrm e}^{-2 \textit {\_a}}+27\right )}+6 \sqrt {3}\, {\mathrm e}^{-2 \textit {\_a}}\right )^{\frac {1}{3}}}{-4 \,{\mathrm e}^{2 \textit {\_a} +3} 3^{\frac {1}{3}} {\left (\left (\sqrt {{\mathrm e}^{-4 \textit {\_a}} \left (4 \,{\mathrm e}^{-2 \textit {\_a}}+27\right )}+3 \sqrt {3}\, {\mathrm e}^{-2 \textit {\_a}}\right )^{2}\right )}^{\frac {1}{3}}-9 i {\mathrm e}^{2 \textit {\_a} +3} 3^{\frac {1}{3}} \left (2 \sqrt {{\mathrm e}^{-4 \textit {\_a}} \left (4 \,{\mathrm e}^{-2 \textit {\_a}}+27\right )}+6 \sqrt {3}\, {\mathrm e}^{-2 \textit {\_a}}\right )^{\frac {1}{3}}-3 \,{\mathrm e}^{2 \textit {\_a} +3} \left (2 \sqrt {{\mathrm e}^{-4 \textit {\_a}} \left (4 \,{\mathrm e}^{-2 \textit {\_a}}+27\right )}+6 \sqrt {3}\, {\mathrm e}^{-2 \textit {\_a}}\right )^{\frac {1}{3}} 3^{\frac {5}{6}}+2 i 2^{\frac {2}{3}} 3^{\frac {5}{6}} {\mathrm e}^{3}-2 \,2^{\frac {2}{3}} 3^{\frac {1}{3}} {\mathrm e}^{3}}d \textit {\_a} \right ) 3^{\frac {5}{6}}+6 i 3^{\frac {1}{3}} \left (\int _{}^{\textit {\_Z}}\frac {{\mathrm e}^{2 \textit {\_a} +3} \left (2 \sqrt {{\mathrm e}^{-4 \textit {\_a}} \left (4 \,{\mathrm e}^{-2 \textit {\_a}}+27\right )}+6 \sqrt {3}\, {\mathrm e}^{-2 \textit {\_a}}\right )^{\frac {1}{3}}}{-4 \,{\mathrm e}^{2 \textit {\_a} +3} 3^{\frac {1}{3}} {\left (\left (\sqrt {{\mathrm e}^{-4 \textit {\_a}} \left (4 \,{\mathrm e}^{-2 \textit {\_a}}+27\right )}+3 \sqrt {3}\, {\mathrm e}^{-2 \textit {\_a}}\right )^{2}\right )}^{\frac {1}{3}}-9 i {\mathrm e}^{2 \textit {\_a} +3} 3^{\frac {1}{3}} \left (2 \sqrt {{\mathrm e}^{-4 \textit {\_a}} \left (4 \,{\mathrm e}^{-2 \textit {\_a}}+27\right )}+6 \sqrt {3}\, {\mathrm e}^{-2 \textit {\_a}}\right )^{\frac {1}{3}}-3 \,{\mathrm e}^{2 \textit {\_a} +3} \left (2 \sqrt {{\mathrm e}^{-4 \textit {\_a}} \left (4 \,{\mathrm e}^{-2 \textit {\_a}}+27\right )}+6 \sqrt {3}\, {\mathrm e}^{-2 \textit {\_a}}\right )^{\frac {1}{3}} 3^{\frac {5}{6}}+2 i 2^{\frac {2}{3}} 3^{\frac {5}{6}} {\mathrm e}^{3}-2 \,2^{\frac {2}{3}} 3^{\frac {1}{3}} {\mathrm e}^{3}}d \textit {\_a} \right )+c_{1} -x \right ) \\ \end{align*} |
Problem 4261
| ODE |
\[ \boxed {{y^{\prime }}^{3}+{\mathrm e}^{-2 y} \left ({\mathrm e}^{2 x}+{\mathrm e}^{3 x}\right ) y^{\prime }-{\mathrm e}^{3 x -2 y}=0} \] |
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program solution |
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Maple solution | \[ y \left (x \right ) = x -\frac {\ln \left (-\frac {{\mathrm e}^{2 x}}{\left (c_{1} +1\right ) \left (-c_{1} +{\mathrm e}^{x}\right )^{2}}\right )}{2} \] |
Problem 4262
| ODE |
\[ \boxed {{y^{\prime }}^{3}+{y^{\prime }}^{2}-y=0} \] |
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program solution |
\[ \int _{}^{y}\frac {6 \left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {1}{3}}}{\left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {2}{3}}-2 \left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {1}{3}}+4}d \textit {\_a} = x +c_{1} \] Verified OK. \[ \int _{}^{y}\frac {12 \left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {2}{3}}-4-4 i \sqrt {3}-\left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {2}{3}}-4 \left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {1}{3}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {12 \left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {2}{3}}+4+\left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {2}{3}}+4 \left (-8+108 \textit {\_a} +12 \sqrt {81 \textit {\_a}^{2}-12 \textit {\_a}}\right )^{\frac {1}{3}}-4 i \sqrt {3}}d \textit {\_a} = x +c_{3} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= 0 \\ 3 \,2^{\frac {1}{3}} \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {\left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {1}{3}}}{\sqrt {3}\, 2^{\frac {1}{3}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {1}{3}}-3^{\frac {1}{3}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {2}{3}}-3^{\frac {2}{3}} 2^{\frac {2}{3}}}d \textit {\_a} \right )+x -c_{1} &= 0 \\ \frac {12 \,2^{\frac {1}{3}} \sqrt {3}\, \left (\int _{}^{y \left (x \right )}-\frac {\left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {1}{3}}}{\left (2^{\frac {1}{3}} 3^{\frac {1}{3}}+3^{\frac {1}{6}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {1}{3}}\right ) \left (i 3^{\frac {5}{6}} 2^{\frac {1}{3}}+2^{\frac {1}{3}} 3^{\frac {1}{3}}-2 \,3^{\frac {1}{6}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {1}{3}}\right )}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (1+i \sqrt {3}\right )}{1+i \sqrt {3}} &= 0 \\ \frac {12 i 2^{\frac {1}{3}} \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {\left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {1}{3}}}{\left (2^{\frac {1}{3}} 3^{\frac {1}{3}}+3^{\frac {1}{6}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {1}{3}}\right ) \left (i 3^{\frac {5}{6}} 2^{\frac {1}{3}}+2 \,3^{\frac {1}{6}} \left (9 \sqrt {27 \textit {\_a}^{2}-4 \textit {\_a}}+\left (27 \textit {\_a} -2\right ) \sqrt {3}\right )^{\frac {1}{3}}-2^{\frac {1}{3}} 3^{\frac {1}{3}}\right )}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (\sqrt {3}+i\right )}{\sqrt {3}+i} &= 0 \\ \end{align*} |
Problem 4263
| ODE |
\[ \boxed {{y^{\prime }}^{3}-{y^{\prime }}^{2}+y^{2}=0} \] |
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program solution |
\[ \int _{}^{y}\frac {6 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{\left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}+2 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}+4}d \textit {\_a} = x +c_{1} \] Verified OK. \[ \int _{}^{y}\frac {12 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{i \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}} \sqrt {3}-4-\left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}+4 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}-4 i \sqrt {3}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {12 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{i \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+4+\left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}-4 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}-4 i \sqrt {3}}d \textit {\_a} = x +c_{3} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= 0 \\ -3 \,3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (\int _{}^{y \left (x \right )}\frac {\left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}+3^{\frac {2}{3}} 2^{\frac {1}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {2}{3}}+6}d \textit {\_a} \right )+x -c_{1} &= 0 \\ \frac {36 \,3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (\int _{}^{y \left (x \right )}\frac {\left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{\left (3 i \sqrt {3}+3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}+3\right ) \left (3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}-6\right )}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (1+i \sqrt {3}\right )}{1+i \sqrt {3}} &= 0 \\ \frac {i \left (x -c_{1} \right ) \sqrt {3}+36 \,3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (\int _{}^{y \left (x \right )}\frac {\left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{\left (-3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}+6\right ) \left (-3 i \sqrt {3}+3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}+3\right )}d \textit {\_a} \right )-x +c_{1}}{-1+i \sqrt {3}} &= 0 \\ \end{align*} |
Problem 4264
| ODE |
\[ \boxed {{y^{\prime }}^{3}-{y^{\prime }}^{2}+x y^{\prime }-y=0} \] |
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program solution |
\[ y = c_{1}^{3}-c_{1}^{2}+c_{1} x \] Verified OK. \[ y = \frac {\left (6 x -2\right ) \sqrt {-3 x +1}}{27}+\frac {x}{3}-\frac {2}{27} \] Verified OK. \[ y = \frac {\left (-6 x +2\right ) \sqrt {-3 x +1}}{27}+\frac {x}{3}-\frac {2}{27} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {x}{3}-\frac {2}{27}-\frac {2 \sqrt {-\left (3 x -1\right )^{3}}}{27} \\ y \left (x \right ) &= \frac {x}{3}-\frac {2}{27}+\frac {2 \sqrt {-\left (3 x -1\right )^{3}}}{27} \\ y \left (x \right ) &= c_{1} \left (c_{1}^{2}-c_{1} +x \right ) \\ \end{align*} |
Problem 4265
| ODE |
\[ \boxed {{y^{\prime }}^{3}-a {y^{\prime }}^{2}+b y=-a b x} \] |
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program solution |
\[ y = \frac {a \left (2 a^{2}-b x \right )}{b} \] Verified OK. \[ x = \frac {-30 a^{2} \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} \ln \left (\frac {\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+8 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a +4 a^{2}}{\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}}}\right )+\left (30 \ln \left (2\right ) a^{2}+30 \ln \left (3\right ) a^{2}+7 a^{2}+6 c_{2} b \right ) \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+\left (14 a^{3}+27 a b x -3 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}+27 b y\right ) \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}}+28 a^{4}-432 a^{2} b x +48 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}\, a -432 y a b}{6 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} b} \] Warning, solution could not be verified \[ x = \frac {112 i \sqrt {3}\, a^{4}-1728 i \sqrt {3}\, a^{2} b x +i \sqrt {3}\, \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {4}{3}}-64 i \sqrt {3}\, \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}+240 a^{2} \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} \ln \left (12\right )+576 i \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}\, a -1728 i \sqrt {3}\, a b y-240 a^{2} \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} \ln \left (\frac {i \left (\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}-4 a^{2}\right ) \sqrt {3}-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+16 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a -4 a^{2}}{\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}}}\right )+\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {4}{3}}+48 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} b c_{2} +56 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} a^{2}-64 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}-112 a^{4}+1728 a^{2} b x -192 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}\, a +1728 y a b}{48 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} b} \] Warning, solution could not be verified \[ x = \frac {-576 a \left (i+\frac {\sqrt {3}}{3}\right ) \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}+i \left (-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {4}{3}}+64 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}-112 a^{4}+1728 a^{2} b x +1728 y a b \right ) \sqrt {3}-112 a^{4}-64 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a^{3}+8 \left (216 b x -\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} \left (30 \ln \left (\frac {i \left (-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+4 a^{2}\right ) \sqrt {3}-\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+16 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}} a -4 a^{2}}{12 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {1}{3}}}\right )-7\right )\right ) a^{2}+1728 y a b +\left (\left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}+48 c_{2} b \right ) \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}}}{48 \left (8 a^{3}-108 a b x +12 \sqrt {3}\, \sqrt {-4 \left (a^{3}-\frac {27 a b x}{4}-\frac {27 b y}{4}\right ) \left (a x +y\right ) b}-108 b y\right )^{\frac {2}{3}} b} \] Warning, solution could not be verified |
|
Maple solution |
\[ y \left (x \right ) = \frac {2 a^{3}-5 \,{\mathrm e}^{\operatorname {RootOf}\left (-10 \textit {\_Z} \,a^{2}-3 \,{\mathrm e}^{2 \textit {\_Z}}+16 a \,{\mathrm e}^{\textit {\_Z}}+2 c_{1} b -13 a^{2}-2 b x \right )} a^{2}+4 \,{\mathrm e}^{2 \operatorname {RootOf}\left (-10 \textit {\_Z} \,a^{2}-3 \,{\mathrm e}^{2 \textit {\_Z}}+16 a \,{\mathrm e}^{\textit {\_Z}}+2 c_{1} b -13 a^{2}-2 b x \right )} a -{\mathrm e}^{3 \operatorname {RootOf}\left (-10 \textit {\_Z} \,a^{2}-3 \,{\mathrm e}^{2 \textit {\_Z}}+16 a \,{\mathrm e}^{\textit {\_Z}}+2 c_{1} b -13 a^{2}-2 b x \right )}-a b x}{b} \] |
Problem 4266
| ODE |
\[ \boxed {{y^{\prime }}^{3}+\operatorname {a0} {y^{\prime }}^{2}+\operatorname {a1} y^{\prime }+\operatorname {a3} y=-\operatorname {a2}} \] |
|
program solution |
\[ \int _{}^{y}\frac {6 \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-2 \operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}+4 \operatorname {a0}^{2}-12 \operatorname {a1}}d \textit {\_a} = x +c_{1} \] Verified OK. \[ \int _{}^{y}\frac {12 \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, \operatorname {a0}^{2}+12 i \sqrt {3}\, \operatorname {a1} -\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-4 \operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}-4 \operatorname {a0}^{2}+12 \operatorname {a1}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {12 \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, \operatorname {a0}^{2}+12 i \sqrt {3}\, \operatorname {a1} +\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}+4 \operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}+4 \operatorname {a0}^{2}-12 \operatorname {a1}}d \textit {\_a} = x +c_{3} \] Verified OK.
|
|
Maple solution |
\begin{align*} x -6 \left (\int _{}^{y \left (x \right )}\frac {\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-2 \operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}+4 \operatorname {a0}^{2}-12 \operatorname {a1}}d \textit {\_a} \right )-c_{1} &= 0 \\ \frac {-12 \left (\int _{}^{y \left (x \right )}\frac {\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{i \left (\operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}+2 \operatorname {a0}^{2}-6 \operatorname {a1} \right ) \sqrt {3}-\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-\operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}+2 \operatorname {a0}^{2}-6 \operatorname {a1}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+x -c_{1}}{1+i \sqrt {3}} &= 0 \\ \frac {-12 \left (\int _{}^{y \left (x \right )}\frac {\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{i \left (\operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}+2 \operatorname {a0}^{2}-6 \operatorname {a1} \right ) \sqrt {3}+\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}+\operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}-2 \operatorname {a0}^{2}+6 \operatorname {a1}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}-x +c_{1}}{-1+i \sqrt {3}} &= 0 \\ \end{align*} |
Problem 4267
| ODE |
\[ \boxed {{y^{\prime }}^{3}+\left (-3 x +1\right ) {y^{\prime }}^{2}-x \left (-3 x +1\right ) y^{\prime }=x^{3}+1} \] |
|
program solution |
\[ y = \int \frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}}+6 x \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}-2 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}-12 x +4}{6 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}d x +c_{1} \] Verified OK. \[ y = \int \frac {i \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}} \sqrt {3}+12 i \sqrt {3}\, x -4 i \sqrt {3}-\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}}+12 x \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}-4 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}+12 x -4}{12 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}d x +c_{2} \] Verified OK. \[ y = \int -\frac {i \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}} \sqrt {3}+12 i \sqrt {3}\, x -4 i \sqrt {3}+\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}}-12 x \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}+4 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}-12 x +4}{12 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}d x +c_{3} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -\frac {\left (\int \frac {\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {2}{3}}+12 \left (x -\frac {1}{3}\right ) \left (i \sqrt {3}-\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {1}{3}}-1\right )}{\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {1}{3}}}d x \right )}{12}+c_{1} \\ y \left (x \right ) &= \frac {\left (\int \frac {\left (-1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {2}{3}}+12 \left (x -\frac {1}{3}\right ) \left (i \sqrt {3}+\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {1}{3}}+1\right )}{\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {1}{3}}}d x \right )}{12}+c_{1} \\ y \left (x \right ) &= \frac {\left (\int \frac {4+6 \left (-2+\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {1}{3}}\right ) x +\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {2}{3}}-2 \left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {1}{3}}}{\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {1}{3}}}d x \right )}{6}+c_{1} \\ \end{align*} |
Problem 4268
| ODE |
\[ \boxed {{y^{\prime }}^{3}-y {y^{\prime }}^{2}+y^{2}=0} \] |
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program solution |
\[ \int _{}^{y}\frac {6 \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}+2 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+4 \textit {\_a}^{2}}d \textit {\_a} = x +c_{1} \] Verified OK. \[ \int _{}^{y}-\frac {12 \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{4 i \textit {\_a}^{2} \sqrt {3}-i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}+4 \textit {\_a}^{2}-4 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}\frac {12 \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{4 i \textit {\_a}^{2} \sqrt {3}+4 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}-4 \textit {\_a}^{2}-i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}-\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{3} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= 0 \\ x -6 \left (\int _{}^{y \left (x \right )}\frac {\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {2}{3}}+2 \left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}} \textit {\_a} +4 \textit {\_a}^{2}}d \textit {\_a} \right )-c_{1} &= 0 \\ \frac {12 \left (\int _{}^{y \left (x \right )}\frac {\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{\left (i \textit {\_a} \sqrt {3}+\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+\textit {\_a} \right ) \left (\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}-2 \textit {\_a} \right )}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+x -c_{1}}{1+i \sqrt {3}} &= 0 \\ \frac {12 \left (\int _{}^{y \left (x \right )}\frac {\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{\left (-i \textit {\_a} \sqrt {3}+\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+\textit {\_a} \right ) \left (-\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+2 \textit {\_a} \right )}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}-x +c_{1}}{-1+i \sqrt {3}} &= 0 \\ \end{align*} |
Problem 4269
| ODE |
\[ \boxed {{y^{\prime }}^{3}+\left (\cos \left (x \right ) \cot \left (x \right )-y\right ) {y^{\prime }}^{2}-\left (1+y \cos \left (x \right ) \cot \left (x \right )\right ) y^{\prime }+y=0} \] |
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program solution |
\[ y = -\cos \left (x \right )+c_{2} \] Verified OK. \[ y = -\ln \left (\tan \left (\frac {x}{2}\right )\right )+c_{3} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= c_{1} {\mathrm e}^{x} \\ y \left (x \right ) &= -\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+c_{1} \\ y \left (x \right ) &= -\cos \left (x \right )+c_{1} \\ \end{align*} |
Problem 4270
| ODE |
\[ \boxed {{y^{\prime }}^{3}+\left (2 x -y^{2}\right ) {y^{\prime }}^{2}-2 x y^{2} y^{\prime }=0} \] |
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program solution |
\[ y = -x^{2}+c_{3} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {1}{c_{1} -x} \\ y \left (x \right ) &= -x^{2}+c_{1} \\ y \left (x \right ) &= c_{1} \\ \end{align*} |
Problem 4271
| ODE |
\[ \boxed {{y^{\prime }}^{3}-\left (y^{2}+2 x \right ) {y^{\prime }}^{2}+\left (x^{2}-y^{2}+2 x y^{2}\right ) y^{\prime }-\left (x^{2}-y^{2}\right ) y^{2}=0} \] |
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program solution |
\[ y = -x -1+c_{3} {\mathrm e}^{x} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {1}{c_{1} -x} \\ y \left (x \right ) &= -x -1+c_{1} {\mathrm e}^{x} \\ y \left (x \right ) &= x -1+{\mathrm e}^{-x} c_{1} \\ \end{align*} |
Problem 4272
| ODE |
\[ \boxed {{y^{\prime }}^{3}-\left (x^{2}+y x +y^{2}\right ) {y^{\prime }}^{2}+x y \left (x^{2}+y x +y^{2}\right ) y^{\prime }-x^{3} y^{3}=0} \] |
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program solution |
\[ y = c_{3} {\mathrm e}^{\frac {x^{2}}{2}} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {x^{3}}{3}+c_{1} \\ y \left (x \right ) &= \frac {1}{c_{1} -x} \\ y \left (x \right ) &= c_{1} {\mathrm e}^{\frac {x^{2}}{2}} \\ \end{align*} |
Problem 4273
| ODE |
\[ \boxed {{y^{\prime }}^{3}-\left (x^{2}+x y^{2}+y^{4}\right ) {y^{\prime }}^{2}+x y^{2} \left (x^{2}+x y^{2}+y^{4}\right ) y^{\prime }-x^{3} y^{6}=0} \] |
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program solution |
\[ y = -\frac {2}{x^{2}+2 c_{3}} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {x^{3}}{3}+c_{1} \\ y \left (x \right ) &= \frac {1}{\left (-3 x +c_{1} \right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {1+i \sqrt {3}}{2 \left (-3 x +c_{1} \right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {-1+i \sqrt {3}}{2 \left (-3 x +c_{1} \right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {2}{x^{2}-2 c_{1}} \\ \end{align*} |
Problem 4274
| ODE |
\[ \boxed {2 {y^{\prime }}^{3}+x y^{\prime }-2 y=0} \] |
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program solution |
\[ y = 0 \] Verified OK. \[ x = \frac {\left (\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}-6 x \right ) \left (\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}+c_{1} \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {1}{3}}-6 x \right )}{6 \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}} \] Verified OK. \[ x = \frac {\left (i \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+6 i \sqrt {3}\, x -\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}+6 x \right ) \left (i \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+6 i \sqrt {3}\, x -\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}+2 c_{1} \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {1}{3}}+6 x \right )}{24 \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}} \] Warning, solution could not be verified \[ x = \frac {\left (i \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+6 i \sqrt {3}\, x +\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}-6 x \right ) \left (i \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}} \sqrt {3}+6 i \sqrt {3}\, x +\left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}-2 c_{1} \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {1}{3}}-6 x \right )}{24 \left (108 y+6 \sqrt {6 x^{3}+324 y^{2}}\right )^{\frac {2}{3}}} \] Warning, solution could not be verified |
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {\left (-c_{1}^{2}-24 x \right ) \sqrt {c_{1}^{2}+24 x}}{432}-\frac {c_{1}^{3}}{432}-\frac {c_{1} x}{12} \\ y \left (x \right ) &= \frac {\left (c_{1}^{2}+24 x \right )^{\frac {3}{2}}}{432}-\frac {c_{1}^{3}}{432}-\frac {c_{1} x}{12} \\ \end{align*} |
Problem 4275
| ODE |
\[ \boxed {2 {y^{\prime }}^{3}+{y^{\prime }}^{2}-y=0} \] |
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program solution |
\[ \int _{}^{y}\frac {6 \left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {1}{3}}}{\left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {2}{3}}-\left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {1}{3}}+1}d \textit {\_a} = x +c_{1} \] Verified OK. \[ \int _{}^{y}\frac {12 \left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {1}{3}}}{i \left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {2}{3}} \sqrt {3}-1-\left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {2}{3}}-2 \left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {1}{3}}-i \sqrt {3}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {12 \left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {1}{3}}}{i \left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {2}{3}} \sqrt {3}+1+\left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {2}{3}}+2 \left (-1+54 \textit {\_a} +6 \sqrt {81 \textit {\_a}^{2}-3 \textit {\_a}}\right )^{\frac {1}{3}}-i \sqrt {3}}d \textit {\_a} = x +c_{3} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= 0 \\ 6 \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {\left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{\frac {1}{3}}}{-3^{\frac {1}{3}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{\frac {2}{3}}+\sqrt {3}\, \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{\frac {1}{3}}-3^{\frac {2}{3}}}d \textit {\_a} \right )+x -c_{1} &= 0 \\ \frac {24 i \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {\left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{\frac {1}{3}}}{\left (3^{\frac {1}{3}}+3^{\frac {1}{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{\frac {1}{3}}\right ) \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}-2 \,3^{\frac {1}{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{\frac {1}{3}}\right )}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (-i+\sqrt {3}\right )}{-i+\sqrt {3}} &= 0 \\ \frac {24 i \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {\left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{\frac {1}{3}}}{\left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}+2 \,3^{\frac {1}{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{\frac {1}{3}}\right ) \left (3^{\frac {1}{3}}+3^{\frac {1}{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{\frac {1}{3}}\right )}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (\sqrt {3}+i\right )}{\sqrt {3}+i} &= 0 \\ \end{align*} |
Problem 4276
| ODE |
\[ \boxed {3 {y^{\prime }}^{3}-x^{4} y^{\prime }+2 x^{3} y=0} \] |
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program solution |
\[ \ln \left (x \right ) = \int _{}^{\frac {y}{x^{3}}}\frac {3 \left (\sqrt {81 \textit {\_a}^{2}-1}-9 \textit {\_a} \right )^{\frac {1}{3}}}{\left (\sqrt {81 \textit {\_a}^{2}-1}-9 \textit {\_a} \right )^{\frac {2}{3}}-9 \left (\sqrt {81 \textit {\_a}^{2}-1}-9 \textit {\_a} \right )^{\frac {1}{3}} \textit {\_a} +1}d \textit {\_a} +c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {x^{3}}{9} \\ y \left (x \right ) &= \frac {x^{3}}{9} \\ y \left (x \right ) &= \frac {c_{1}^{2} x^{2}-3}{2 c_{1}^{3}} \\ \end{align*} |
Problem 4277
| ODE |
\[ \boxed {4 {y^{\prime }}^{3}+4 y^{\prime }=x} \] |
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program solution |
\[ y = \int \frac {\left (27 x +3 \sqrt {81 x^{2}+192}\right )^{\frac {2}{3}}-12}{6 \left (27 x +3 \sqrt {81 x^{2}+192}\right )^{\frac {1}{3}}}d x +c_{1} \] Verified OK. \[ y = \int \frac {i \sqrt {3}\, \left (27 x +3 \sqrt {81 x^{2}+192}\right )^{\frac {2}{3}}+12 i \sqrt {3}-\left (27 x +3 \sqrt {81 x^{2}+192}\right )^{\frac {2}{3}}+12}{12 \left (27 x +3 \sqrt {81 x^{2}+192}\right )^{\frac {1}{3}}}d x +c_{2} \] Verified OK. \[ y = \int -\frac {i \sqrt {3}\, \left (27 x +3 \sqrt {81 x^{2}+192}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (27 x +3 \sqrt {81 x^{2}+192}\right )^{\frac {2}{3}}-12}{12 \left (27 x +3 \sqrt {81 x^{2}+192}\right )^{\frac {1}{3}}}d x +c_{3} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {\left (\int \frac {\left (-1+i \sqrt {3}\right ) \left (27 x +3 \sqrt {81 x^{2}+192}\right )^{\frac {2}{3}}+12 i \sqrt {3}+12}{\left (27 x +3 \sqrt {81 x^{2}+192}\right )^{\frac {1}{3}}}d x \right )}{12}+c_{1} \\ y \left (x \right ) &= -\frac {\left (\int \frac {i \sqrt {3}\, \left (27 x +3 \sqrt {81 x^{2}+192}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (27 x +3 \sqrt {81 x^{2}+192}\right )^{\frac {2}{3}}-12}{\left (27 x +3 \sqrt {81 x^{2}+192}\right )^{\frac {1}{3}}}d x \right )}{12}+c_{1} \\ y \left (x \right ) &= \frac {\left (\int \frac {\left (27 x +3 \sqrt {81 x^{2}+192}\right )^{\frac {2}{3}}-12}{\left (27 x +3 \sqrt {81 x^{2}+192}\right )^{\frac {1}{3}}}d x \right )}{6}+c_{1} \\ \end{align*} |
Problem 4278
| ODE |
\[ \boxed {8 {y^{\prime }}^{3}+12 {y^{\prime }}^{2}-27 y=27 x} \] |
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program solution |
\[ y = \frac {4}{27}-x \] Verified OK. \[ y = -\left (x +c_{1} \right )^{\frac {3}{2}}+c_{1} \] Verified OK. \[ y = \left (x +c_{1} \right )^{\frac {3}{2}}+c_{1} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -x +\frac {4}{27} \\ y \left (x \right ) &= \left (c_{1} -x \right ) \sqrt {x -c_{1}}-c_{1} \\ y \left (x \right ) &= \left (x -c_{1} \right )^{\frac {3}{2}}-c_{1} \\ \end{align*} |
Problem 4279
| ODE |
\[ \boxed {x {y^{\prime }}^{3}-y {y^{\prime }}^{2}=-a} \] |
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program solution |
\[ y = c_{1} x +\frac {a}{c_{1}^{2}} \] Verified OK. \[ y = \frac {3 a \,x^{2} 2^{\frac {1}{3}}}{2 \left (a \,x^{2}\right )^{\frac {2}{3}}} \] Verified OK. \[ y = -\frac {3 a \,x^{2} 2^{\frac {1}{3}}}{\left (a \,x^{2}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )} \] Verified OK. \[ y = \frac {3 a \,x^{2} 2^{\frac {1}{3}}}{\left (a \,x^{2}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {3 \,2^{\frac {1}{3}} \left (a \,x^{2}\right )^{\frac {1}{3}}}{2} \\ y \left (x \right ) &= -\frac {3 \,2^{\frac {1}{3}} \left (a \,x^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {3 \,2^{\frac {1}{3}} \left (a \,x^{2}\right )^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {c_{1}^{3} x +a}{c_{1}^{2}} \\ \end{align*} |
Problem 4280
| ODE |
\[ \boxed {x {y^{\prime }}^{3}-\left (x +x^{2}+y\right ) {y^{\prime }}^{2}+\left (x^{2}+y+y x \right ) y^{\prime }-y x=0} \] |
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program solution |
\[ y = c_{3} x \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= c_{1} x \\ y \left (x \right ) &= x +c_{1} \\ y \left (x \right ) &= \frac {x^{2}}{2}+c_{1} \\ \end{align*} |
Problem 4281
| ODE |
\[ \boxed {x {y^{\prime }}^{3}-2 y {y^{\prime }}^{2}=-4 x^{2}} \] |
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program solution |
\[ \frac {4 \ln \left (x \right )}{3}-4 \left (\int _{}^{\frac {y}{x^{\frac {4}{3}}}}\frac {\left (-54+8 \textit {\_a}^{3}+6 i \sqrt {24 \textit {\_a}^{3}-81}\right )^{\frac {1}{3}}}{4 \textit {\_a}^{2}-2 \textit {\_a} \left (-54+8 \textit {\_a}^{3}+6 i \sqrt {24 \textit {\_a}^{3}-81}\right )^{\frac {1}{3}}+\left (-54+8 \textit {\_a}^{3}+6 i \sqrt {24 \textit {\_a}^{3}-81}\right )^{\frac {2}{3}}}d \textit {\_a} \right )-c_{1} = 0 \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {3 x^{\frac {4}{3}}}{2} \\ y \left (x \right ) &= -\frac {3 x^{\frac {4}{3}} \left (1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {3 x^{\frac {4}{3}} \left (-1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {c_{1}^{3}+128 x^{2}}{32 c_{1}} \\ y \left (x \right ) &= \frac {c_{1}^{3}-128 x^{2}}{32 c_{1}} \\ y \left (x \right ) &= \frac {c_{1} \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {-x^{2} \left (c_{1}^{3}-864 x^{2}\right )}\right )^{\frac {1}{3}}}{96}+\frac {c_{1}^{3}}{96 \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {-x^{2} \left (c_{1}^{3}-864 x^{2}\right )}\right )^{\frac {1}{3}}}+\frac {c_{1}^{2}}{96} \\ y \left (x \right ) &= \frac {c_{1} \left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {x^{2} \left (c_{1}^{3}+864 x^{2}\right )}+1728 x^{2}\right )^{\frac {1}{3}}}{96}+\frac {c_{1}^{3}}{96 \left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {x^{2} \left (c_{1}^{3}+864 x^{2}\right )}+1728 x^{2}\right )^{\frac {1}{3}}}+\frac {c_{1}^{2}}{96} \\ y \left (x \right ) &= \frac {\left (c_{1} -\left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}\right ) c_{1} \left (i \left (\left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}+c_{1} \right ) \sqrt {3}-c_{1} +\left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}\right )}{192 \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {6}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (-1+i \sqrt {3}\right ) c_{1} \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {3}\, \sqrt {2}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}}{192}-\frac {\left (i \sqrt {3}\, c_{1} +c_{1} -2 \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {3}\, \sqrt {2}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}\right ) c_{1}^{2}}{192 \left (-1728 x^{2}+c_{1}^{3}+24 \sqrt {3}\, \sqrt {2}\, \sqrt {-c_{1}^{3} x^{2}+864 x^{4}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (c_{1} -\left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}\right ) c_{1} \left (i \left (\left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}+c_{1} \right ) \sqrt {3}-c_{1} +\left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}\right )}{192 \left (c_{1}^{3}+24 \sqrt {6}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (-1+i \sqrt {3}\right ) c_{1} \left (c_{1}^{3}+24 \sqrt {3}\, \sqrt {2}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}}{192}-\frac {\left (i \sqrt {3}\, c_{1} +c_{1} -2 \left (c_{1}^{3}+24 \sqrt {3}\, \sqrt {2}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}\right ) c_{1}^{2}}{192 \left (c_{1}^{3}+24 \sqrt {3}\, \sqrt {2}\, \sqrt {c_{1}^{3} x^{2}+864 x^{4}}+1728 x^{2}\right )^{\frac {1}{3}}} \\ \end{align*} |
Problem 4282
| ODE |
\[ \boxed {2 x {y^{\prime }}^{3}-3 y {y^{\prime }}^{2}=x} \] |
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program solution |
\[ y = -x \] Verified OK. \[ y = -\frac {4 x}{-2 i \sqrt {3}-2} \] Verified OK. \[ y = -\frac {4 x}{-2+2 i \sqrt {3}} \] Verified OK. \[ y = \frac {2 c_{1}^{3} x^{\frac {3}{2}}-1}{3 c_{1}^{2}} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -\frac {\left (-1+i \sqrt {3}\right ) x}{2} \\ y \left (x \right ) &= \frac {\left (1+i \sqrt {3}\right ) x}{2} \\ y \left (x \right ) &= -x \\ y \left (x \right ) &= \frac {2 x \sqrt {c_{1} x}-c_{1}^{2}}{3 c_{1}} \\ y \left (x \right ) &= \frac {-c_{1}^{2}-2 x \sqrt {c_{1} x}}{3 c_{1}} \\ \end{align*} |
Problem 4283
| ODE |
\[ \boxed {4 x {y^{\prime }}^{3}-6 y {y^{\prime }}^{2}+3 y=x} \] |
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program solution |
\[ y = x \] Verified OK. \[ y = \frac {-x \sqrt {3}-2 x}{1+\sqrt {3}} \] Verified OK. \[ y = \frac {-x \sqrt {3}+2 x}{\sqrt {3}-1} \] Verified OK. \[ x = c_{3} \left (-1+\frac {{\left (\left (y^{3}-3 y x^{2}+x^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )^{\frac {1}{3}}+\frac {y^{2}}{\left (y^{3}-3 y x^{2}+x^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )^{\frac {1}{3}}}+y\right )}^{2}}{2 x^{2}}\right ) \] Verified OK. \[ x = \frac {\left (\left (-2 x^{2}+3 y^{2}\right ) \left (y^{3}-3 y x^{2}+x^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )^{\frac {2}{3}}-\frac {\left (1+i \sqrt {3}\right ) \left (x^{3}-3 y x^{2}+3 y^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right ) \left (y^{3}-3 y x^{2}+x^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )^{\frac {1}{3}}}{2}+\left (i \sqrt {3}-1\right ) y \left (x^{3}-3 y x^{2}+\frac {3 y^{3}}{2}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )\right ) c_{3}}{2 \left (y^{3}-3 y x^{2}+x^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )^{\frac {2}{3}} x^{2}} \] Verified OK. \[ x = -\frac {\left (\left (2 x^{2}-3 y^{2}\right ) \left (y^{3}-3 y x^{2}+x^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )^{\frac {2}{3}}-\frac {\left (i \sqrt {3}-1\right ) \left (x^{3}-3 y x^{2}+3 y^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right ) \left (y^{3}-3 y x^{2}+x^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )^{\frac {1}{3}}}{2}+\left (1+i \sqrt {3}\right ) y \left (x^{3}-3 y x^{2}+\frac {3 y^{3}}{2}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )\right ) c_{3}}{2 \left (y^{3}-3 y x^{2}+x^{3}+\sqrt {-6 y^{4}+2 x y^{3}+9 y^{2} x^{2}-6 x^{3} y+x^{4}}\, x \right )^{\frac {2}{3}} x^{2}} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -\frac {\left (1+\sqrt {3}\right ) x}{2} \\ y \left (x \right ) &= \frac {\left (\sqrt {3}-1\right ) x}{2} \\ y \left (x \right ) &= x \\ y \left (x \right ) &= \frac {-\left (x +c_{1} \right ) \sqrt {2}\, \sqrt {c_{1} \left (x +c_{1} \right )}-c_{1}^{2}}{3 c_{1}} \\ y \left (x \right ) &= \frac {\left (x +c_{1} \right ) \sqrt {2}\, \sqrt {c_{1} \left (x +c_{1} \right )}-c_{1}^{2}}{3 c_{1}} \\ \end{align*} |
Problem 4284
| ODE |
\[ \boxed {8 x {y^{\prime }}^{3}-12 y {y^{\prime }}^{2}+9 y=0} \] |
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program solution |
\[ y = 0 \] Verified OK. \[ y = -\frac {3 x}{2} \] Verified OK. \[ y = \frac {3 x}{2} \] Verified OK. \[ x = c_{3} \left (\frac {\left (2^{\frac {2}{3}} {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {2}{3}}+2 y^{2} 2^{\frac {1}{3}}+2 y {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {1}{3}}\right )^{2}}{4 x^{2} {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {2}{3}}}-3\right ) \] Verified OK. \[ x = c_{3} \left (-3+\frac {\left (-i 2^{\frac {2}{3}} {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {2}{3}} \sqrt {3}+2 i 2^{\frac {1}{3}} \sqrt {3}\, y^{2}+2^{\frac {2}{3}} {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {2}{3}}+2 y^{2} 2^{\frac {1}{3}}-4 y {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {1}{3}}\right )^{2}}{16 x^{2} {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {2}{3}}}\right ) \] Verified OK. \[ x = -c_{3} \left (\frac {\left (-2^{\frac {2}{3}} {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {2}{3}} \sqrt {3}+2 \,2^{\frac {1}{3}} \sqrt {3}\, y^{2}+i 2^{\frac {2}{3}} {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {2}{3}}+2 i y^{2} 2^{\frac {1}{3}}-4 i y {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {1}{3}}\right )^{2}}{16 x^{2} {\left (y \left (2 y^{2}+3 \sqrt {-4 y^{2}+9 x^{2}}\, x -9 x^{2}\right )\right )}^{\frac {2}{3}}}+3\right ) \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -\frac {3 x}{2} \\ y \left (x \right ) &= \frac {3 x}{2} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {\left (3 c_{1} +x \right ) \sqrt {c_{1} \left (3 c_{1} +x \right )}}{3 c_{1}} \\ y \left (x \right ) &= \frac {\left (3 c_{1} +x \right ) \sqrt {c_{1} \left (3 c_{1} +x \right )}}{3 c_{1}} \\ \end{align*} |
Problem 4285
| ODE |
\[ \boxed {x^{2} {y^{\prime }}^{3}-2 x y {y^{\prime }}^{2}+y^{2} y^{\prime }=-1} \] |
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program solution |
\[ y = c_{1} x -\frac {1}{\sqrt {-c_{1}}} \] Verified OK. \[ y = -\frac {\left (\frac {\left (x^{2}\right )^{\frac {2}{3}} \sqrt {\frac {\left (x^{2}\right )^{\frac {2}{3}}}{x^{2}}}}{2}+x \right ) 2^{\frac {1}{3}}}{\sqrt {\frac {\left (x^{2}\right )^{\frac {2}{3}}}{x^{2}}}\, x} \] Verified OK. \[ y = \frac {\left (1+i \sqrt {3}\right ) \left (x^{2}\right )^{\frac {2}{3}} 2^{\frac {1}{3}} \sqrt {-\frac {\left (x^{2}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )}{x^{2}}}-4 \,2^{\frac {5}{6}} x}{4 \sqrt {-\frac {\left (x^{2}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )}{x^{2}}}\, x} \] Verified OK. \[ y = -\frac {\left (x^{2}\right )^{\frac {2}{3}} 2^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) \sqrt {\frac {\left (x^{2}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )}{x^{2}}}+4 \,2^{\frac {5}{6}} x}{4 \sqrt {\frac {\left (x^{2}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )}{x^{2}}}\, x} \] Verified OK. \[ y = c_{2} x +\frac {1}{\sqrt {-c_{2}}} \] Verified OK. \[ y = -\frac {2^{\frac {1}{3}} \left (\left (-x^{2}\right )^{\frac {2}{3}} \sqrt {\frac {\left (-x^{2}\right )^{\frac {2}{3}}}{x^{2}}}-2 x \right )}{2 x \sqrt {\frac {\left (-x^{2}\right )^{\frac {2}{3}}}{x^{2}}}} \] Verified OK. \[ y = -\frac {2^{\frac {5}{6}} \left (\left (i \sqrt {6}-\sqrt {2}\right ) \left (-x^{2}\right )^{\frac {2}{3}} \sqrt {\frac {\left (-x^{2}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )}{x^{2}}}-8 x \right )}{8 \sqrt {\frac {\left (-x^{2}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )}{x^{2}}}\, x} \] Verified OK. \[ y = \frac {\left (\left (i \sqrt {6}+\sqrt {2}\right ) \left (-x^{2}\right )^{\frac {2}{3}} \sqrt {-\frac {\left (-x^{2}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )}{x^{2}}}+8 x \right ) 2^{\frac {5}{6}}}{8 \sqrt {-\frac {\left (-x^{2}\right )^{\frac {2}{3}} \left (1+i \sqrt {3}\right )}{x^{2}}}\, x} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {3 \,2^{\frac {1}{3}} \left (-x \right )^{\frac {1}{3}}}{2} \\ y \left (x \right ) &= -\frac {3 \,2^{\frac {1}{3}} \left (-x \right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {3 \,2^{\frac {1}{3}} \left (-x \right )^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= c_{1} x -\frac {1}{\sqrt {-c_{1}}} \\ y \left (x \right ) &= c_{1} x +\frac {1}{\sqrt {-c_{1}}} \\ \end{align*} |
Problem 4286
| ODE |
\[ \boxed {\left (a^{2}-x^{2}\right ) {y^{\prime }}^{3}+b x \left (a^{2}-x^{2}\right ) {y^{\prime }}^{2}-y^{\prime }=b x} \] |
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program solution |
\[ y = -\frac {b \,x^{2}}{2}+c_{3} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {b \,x^{2}}{2}+c_{1} \\ y \left (x \right ) &= \arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_{1} \\ y \left (x \right ) &= -\arctan \left (\frac {x}{\sqrt {a^{2}-x^{2}}}\right )+c_{1} \\ \end{align*} |
Problem 4287
| ODE |
\[ \boxed {x {y^{\prime }}^{3}-3 x^{2} y {y^{\prime }}^{2}+x \left (x^{5}+3 y^{2}\right ) y^{\prime }-2 x^{5} y-y^{3}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 4288
| ODE |
\[ \boxed {2 x^{3} {y^{\prime }}^{3}+6 x^{2} y {y^{\prime }}^{2}-\left (1-6 y x \right ) y y^{\prime }+2 y^{3}=0} \] |
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program solution |
\[ -\ln \left (x \right ) = \int _{}^{y x}-\frac {\left (3 i+\sqrt {3}\right ) \left (12 \sqrt {3}\, \sqrt {\textit {\_a}}-4 \sqrt {27 \textit {\_a} -2}\right )^{\frac {1}{3}}}{\left (i \left (3 \sqrt {3}\, \sqrt {\textit {\_a}}-\sqrt {27 \textit {\_a} -2}\right )^{\frac {2}{3}} \sqrt {3}-\left (3 \sqrt {3}\, \sqrt {\textit {\_a}}-\sqrt {27 \textit {\_a} -2}\right )^{\frac {2}{3}}+2 \,2^{\frac {1}{3}}\right ) \sqrt {\textit {\_a}}}d \textit {\_a} +c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= 0 \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ \end{align*} |
Problem 4289
| ODE |
\[ \boxed {x^{4} {y^{\prime }}^{3}-x^{3} y {y^{\prime }}^{2}-x^{2} y^{2} y^{\prime }+x y^{3}=1} \] |
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program solution |
\[ -\frac {\ln \left (x \right )}{3} = \int _{}^{y x^{\frac {1}{3}}}-\frac {2 \left (-16 \textit {\_a}^{3}+3 \sqrt {-96 \textit {\_a}^{3}+81}+27\right )^{\frac {1}{3}}}{2^{\frac {2}{3}} \left (-16 \textit {\_a}^{3}+3 \sqrt {-96 \textit {\_a}^{3}+81}+27\right )^{\frac {2}{3}}+8 \,2^{\frac {1}{3}} \textit {\_a}^{2}+4 \left (-16 \textit {\_a}^{3}+3 \sqrt {-96 \textit {\_a}^{3}+81}+27\right )^{\frac {1}{3}} \textit {\_a}}d \textit {\_a} +c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}}}{4 x} \\ y \left (x \right ) &= -\frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{8 x} \\ y \left (x \right ) &= \frac {3 \,2^{\frac {1}{3}} \left (x^{2}\right )^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )}{8 x} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+6 \left (\int _{}^{\textit {\_Z}}\frac {\left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}{8 \,2^{\frac {2}{3}} \textit {\_a}^{2}+2^{\frac {1}{3}} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {2}{3}}+4 \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}d \textit {\_a} \right )+c_{1} \right )}{x^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (3 i \sqrt {3}\, \left (\int _{}^{\textit {\_Z}}\frac {\left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}{4 i \sqrt {3}\, 2^{\frac {2}{3}} \textit {\_a}^{2}-2 i \textit {\_a} \sqrt {3}\, \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}+4 \,2^{\frac {2}{3}} \textit {\_a}^{2}-2^{\frac {1}{3}} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {2}{3}}+2 \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}d \textit {\_a} \right )+\ln \left (x \right )-3 \left (\int _{}^{\textit {\_Z}}\frac {\left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}{4 i \sqrt {3}\, 2^{\frac {2}{3}} \textit {\_a}^{2}-2 i \textit {\_a} \sqrt {3}\, \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}+4 \,2^{\frac {2}{3}} \textit {\_a}^{2}-2^{\frac {1}{3}} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {2}{3}}+2 \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}d \textit {\_a} \right )-c_{1} \right )}{x^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\operatorname {RootOf}\left (3 i \sqrt {3}\, \left (\int _{}^{\textit {\_Z}}\frac {\left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}{4 i \sqrt {3}\, 2^{\frac {2}{3}} \textit {\_a}^{2}-2 i \textit {\_a} \sqrt {3}\, \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}-4 \,2^{\frac {2}{3}} \textit {\_a}^{2}+2^{\frac {1}{3}} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {2}{3}}-2 \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}d \textit {\_a} \right )+\ln \left (x \right )+3 \left (\int _{}^{\textit {\_Z}}\frac {\left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}{4 i \sqrt {3}\, 2^{\frac {2}{3}} \textit {\_a}^{2}-2 i \textit {\_a} \sqrt {3}\, \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}-4 \,2^{\frac {2}{3}} \textit {\_a}^{2}+2^{\frac {1}{3}} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {2}{3}}-2 \textit {\_a} \left (-32 \textit {\_a}^{3}+6 \sqrt {-96 \textit {\_a}^{3}+81}+54\right )^{\frac {1}{3}}}d \textit {\_a} \right )-c_{1} \right )}{x^{\frac {1}{3}}} \\ \end{align*} |
Problem 4290
| ODE |
\[ \boxed {x^{6} {y^{\prime }}^{3}-x y^{\prime }-y=0} \] |
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program solution |
\[ -\frac {3 \ln \left (x \right )}{2} = \int _{}^{y x^{\frac {3}{2}}}\frac {-3 \left (3 \sqrt {3}\, \textit {\_a} +\sqrt {27 \textit {\_a}^{2}-4}\right )^{\frac {2}{3}} \sqrt {3}\, 2^{\frac {1}{3}} \textit {\_a} +\left (3 \sqrt {3}\, \textit {\_a} +\sqrt {27 \textit {\_a}^{2}-4}\right )^{\frac {2}{3}} 2^{\frac {1}{3}} \sqrt {27 \textit {\_a}^{2}-4}+2 \,2^{\frac {2}{3}} \left (3 \sqrt {3}\, \textit {\_a} +\sqrt {27 \textit {\_a}^{2}-4}\right )^{\frac {1}{3}}-2 \sqrt {27 \textit {\_a}^{2}-4}}{2 \textit {\_a} \sqrt {27 \textit {\_a}^{2}-4}}d \textit {\_a} +c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {2 \sqrt {3}}{9 x^{\frac {3}{2}}} \\ y \left (x \right ) &= \frac {2 \sqrt {3}}{9 x^{\frac {3}{2}}} \\ y \left (x \right ) &= c_{1}^{3}-\frac {c_{1}}{x} \\ \end{align*} |
Problem 4291
| ODE |
\[ \boxed {y {y^{\prime }}^{3}-3 x y^{\prime }+3 y=0} \] |
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program solution |
\[ y = 0 \] Verified OK. \[ x = \frac {6 c_{2} y 2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}} x \,{\mathrm e}^{-\frac {4 y^{5} \left (-3 y+\sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )}{\left (2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}}+2 y x \right )^{3}}}}{\left (2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}}+2 y x \right )^{2}} \] Warning, solution could not be verified \[ x = \frac {6 c_{2} y 2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}} x \,{\mathrm e}^{-\frac {32 y^{5} \left (-3 y+\sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )}{\left (i \sqrt {3}\, 2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}}-2 i \sqrt {3}\, x y-2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}}-2 y x \right )^{3}}}}{\left (-\frac {2^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}}}{2}+x \left (1+i \sqrt {3}\right ) y\right )^{2}} \] Warning, solution could not be verified \[ x = \frac {6 c_{2} y 2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}} x \,{\mathrm e}^{\frac {96 y^{6}-32 y^{5} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}}{8 \left (-\frac {\left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}}}{2}+x y \left (i \sqrt {3}-1\right )\right )^{3}}}}{\left (-\frac {\left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-3 y^{3}+y^{2} \sqrt {\frac {9 y^{3}-4 x^{3}}{y}}\right )^{\frac {2}{3}}}{2}+x y \left (i \sqrt {3}-1\right )\right )^{2}} \] Warning, solution could not be verified |
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Maple solution |
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {-2 {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{\frac {1}{3}} \textit {\_a}^{3}-8 \textit {\_a}^{3}+{\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{\frac {2}{3}}+2 {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{\frac {1}{3}}+4}{\textit {\_a}^{4} {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{\frac {1}{3}}}d \textit {\_a} +2 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-4 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {8 i \sqrt {3}\, \textit {\_a}^{3}+i \sqrt {3}\, {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{\frac {2}{3}}-4 {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{\frac {1}{3}} \textit {\_a}^{3}+8 \textit {\_a}^{3}-4 i \sqrt {3}-{\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{\frac {2}{3}}+4 {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{\frac {1}{3}}-4}{\textit {\_a}^{4} {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{\frac {1}{3}}}d \textit {\_a} +4 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-4 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {8 i \sqrt {3}\, \textit {\_a}^{3}+i \sqrt {3}\, {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{\frac {2}{3}}+4 {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{\frac {1}{3}} \textit {\_a}^{3}-8 \textit {\_a}^{3}-4 i \sqrt {3}+{\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{\frac {2}{3}}-4 {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{\frac {1}{3}}+4}{\textit {\_a}^{4} {\left (4 \sqrt {\frac {9 \textit {\_a}^{3}-4}{\textit {\_a}}}\, \textit {\_a}^{5}+12 \textit {\_a}^{6}-24 \textit {\_a}^{3}+8\right )}^{\frac {1}{3}}}d \textit {\_a} \right )+4 c_{1} \right ) x \\ \end{align*} |
Problem 4292
| ODE |
\[ \boxed {2 y {y^{\prime }}^{3}-3 x y^{\prime }+2 y=0} \] |
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program solution |
\[ y = 0 \] Verified OK. \[ y = \frac {2^{\frac {2}{3}} x}{2} \] Verified OK. \[ y = -\frac {i \sqrt {3}\, 2^{\frac {2}{3}} x}{4}-\frac {2^{\frac {2}{3}} x}{4} \] Verified OK. \[ y = \frac {i \sqrt {3}\, 2^{\frac {2}{3}} x}{4}-\frac {2^{\frac {2}{3}} x}{4} \] Verified OK. \[ x = \frac {\left (\left (2 x +2 y\right ) \left (-2 y^{3}+y^{2} \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right )^{\frac {2}{3}}+y \left (-2^{\frac {2}{3}} \left (x +y-\frac {\sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}}{2}\right ) \left (-2 y^{3}+y^{2} \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right )^{\frac {1}{3}}+2^{\frac {1}{3}} \left (x^{2}+2 y^{2}-y \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right )\right )\right ) \left (2^{\frac {1}{3}} \left (-2 y^{3}+y^{2} \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right )^{\frac {2}{3}}+y \left (2^{\frac {2}{3}} x +2 \left (-2 y^{3}+y^{2} \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right )^{\frac {1}{3}}\right )\right ) y c_{3}}{\left (2^{\frac {1}{3}} x y+\left (-2 y^{3}+y^{2} \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right )^{\frac {2}{3}}\right )^{3}} \] Verified OK. \[ x = \frac {2 \left (\left (1-i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-2 y^{3}+y^{2} \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right )^{\frac {2}{3}}+\left (-4 \left (-2 y^{3}+y^{2} \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right )^{\frac {1}{3}}+\left (1+i \sqrt {3}\right ) 2^{\frac {2}{3}} x \right ) y\right ) \left (\left (4 y+4 x \right ) \left (-2 y^{3}+y^{2} \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right )^{\frac {2}{3}}+\left (2^{\frac {2}{3}} \left (x +y-\frac {\sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}}{2}\right ) \left (1+i \sqrt {3}\right ) \left (-2 y^{3}+y^{2} \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right )^{\frac {1}{3}}+\left (x^{2}+2 y^{2}-y \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right ) 2^{\frac {1}{3}} \left (i \sqrt {3}-1\right )\right ) y\right ) y c_{3}}{{\left (\left (1-i \sqrt {3}\right ) \left (-2 y^{3}+y^{2} \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right )^{\frac {2}{3}}+x 2^{\frac {1}{3}} y \left (1+i \sqrt {3}\right )\right )}^{3}} \] Warning, solution could not be verified \[ x = -\frac {2 \left (\left (-1-i \sqrt {3}\right ) 2^{\frac {1}{3}} \left (-2 y^{3}+y^{2} \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right )^{\frac {2}{3}}+\left (4 \left (-2 y^{3}+y^{2} \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) 2^{\frac {2}{3}} x \right ) y\right ) \left (\left (-4 x -4 y\right ) \left (-2 y^{3}+y^{2} \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right )^{\frac {2}{3}}+\left (2^{\frac {2}{3}} \left (x +y-\frac {\sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}}{2}\right ) \left (i \sqrt {3}-1\right ) \left (-2 y^{3}+y^{2} \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right )^{\frac {1}{3}}+\left (x^{2}+2 y^{2}-y \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right ) 2^{\frac {1}{3}} \left (1+i \sqrt {3}\right )\right ) y\right ) y c_{3}}{{\left (\left (-1-i \sqrt {3}\right ) \left (-2 y^{3}+y^{2} \sqrt {\frac {-2 x^{3}+4 y^{3}}{y}}\right )^{\frac {2}{3}}+x 2^{\frac {1}{3}} y \left (i \sqrt {3}-1\right )\right )}^{3}} \] Warning, solution could not be verified |
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {2^{\frac {2}{3}} x}{2} \\ y \left (x \right ) &= -\frac {2^{\frac {2}{3}} \left (1+i \sqrt {3}\right ) x}{4} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} \left (-1+i \sqrt {3}\right ) x}{4} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {2 {\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )}^{\frac {1}{3}} \textit {\_a}^{3}+2 \textit {\_a}^{3}-{\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )}^{\frac {2}{3}}-{\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )}^{\frac {1}{3}}-1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right ) {\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )}^{\frac {1}{3}}}d \textit {\_a} +c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )+\int _{}^{\textit {\_Z}}\frac {2 i \sqrt {3}\, \textit {\_a}^{3}+i \sqrt {3}\, {\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )}^{\frac {2}{3}}-4 {\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )}^{\frac {1}{3}} \textit {\_a}^{3}+2 \textit {\_a}^{3}-i \sqrt {3}-{\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )}^{\frac {2}{3}}+2 {\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )}^{\frac {1}{3}}-1}{{\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )}^{\frac {1}{3}} \textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}d \textit {\_a} +2 c_{1} \right ) x \\ y \left (x \right ) &= \operatorname {RootOf}\left (-2 \ln \left (x \right )-\left (\int _{}^{\textit {\_Z}}\frac {2 i \sqrt {3}\, \textit {\_a}^{3}+i \sqrt {3}\, {\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )}^{\frac {2}{3}}+4 {\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )}^{\frac {1}{3}} \textit {\_a}^{3}-2 \textit {\_a}^{3}-i \sqrt {3}+{\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )}^{\frac {2}{3}}-2 {\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )}^{\frac {1}{3}}+1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right ) {\left (\left (\sqrt {2}\, \sqrt {\frac {1}{\textit {\_a} \left (2 \textit {\_a}^{3}-1\right )}}\, \textit {\_a}^{2}+1\right ) \left (2 \textit {\_a}^{3}-1\right )^{2}\right )}^{\frac {1}{3}}}d \textit {\_a} \right )+2 c_{1} \right ) x \\ \end{align*} |
Problem 4293
| ODE |
\[ \boxed {\left (x +2 y\right ) {y^{\prime }}^{3}+3 \left (y+x \right ) {y^{\prime }}^{2}+\left (2 x +y\right ) y^{\prime }=0} \] |
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program solution |
\[ x^{2}+y x +y^{2} = {\mathrm e}^{c_{4}} c_{5} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= c_{1} -x \\ y \left (x \right ) &= \frac {-c_{1} x -\sqrt {-3 c_{1}^{2} x^{2}+4}}{2 c_{1}} \\ y \left (x \right ) &= \frac {-c_{1} x +\sqrt {-3 c_{1}^{2} x^{2}+4}}{2 c_{1}} \\ y \left (x \right ) &= c_{1} \\ \end{align*} |
Problem 4294
| ODE |
\[ \boxed {y^{2} {y^{\prime }}^{3}-x y^{\prime }+y=0} \] |
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program solution |
\[ -9 \left (\int _{}^{\frac {y}{x^{\frac {3}{4}}}}\frac {\left (3 \sqrt {3}\, \textit {\_a}^{2}-\sqrt {27 \textit {\_a}^{4}-4}\right )^{\frac {1}{3}} \textit {\_a}}{6 \left (i 3^{\frac {1}{6}}-\frac {3^{\frac {2}{3}}}{3}\right ) 2^{\frac {2}{3}} \left (3 \sqrt {3}\, \textit {\_a}^{2}-\sqrt {27 \textit {\_a}^{4}-4}\right )^{\frac {2}{3}}-9 \textit {\_a}^{2} \left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) \left (3 \sqrt {3}\, \textit {\_a}^{2}-\sqrt {27 \textit {\_a}^{4}-4}\right )^{\frac {1}{3}}+8 \,2^{\frac {1}{3}} 3^{\frac {2}{3}}}d \textit {\_a} \right ) \left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right )-c_{1} +\frac {3 \ln \left (x \right )}{4} = 0 \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {2 \sqrt {-24 c_{1}^{3}+27 c_{1} x -3 \sqrt {\left (4 c_{1}^{2}-3 x \right )^{3}}}}{9} \\ y \left (x \right ) &= \frac {2 \sqrt {-24 c_{1}^{3}+27 c_{1} x -3 \sqrt {\left (4 c_{1}^{2}-3 x \right )^{3}}}}{9} \\ y \left (x \right ) &= -\frac {2 \sqrt {-24 c_{1}^{3}+27 c_{1} x +3 \sqrt {\left (4 c_{1}^{2}-3 x \right )^{3}}}}{9} \\ y \left (x \right ) &= \frac {2 \sqrt {-24 c_{1}^{3}+27 c_{1} x +3 \sqrt {\left (4 c_{1}^{2}-3 x \right )^{3}}}}{9} \\ \end{align*} |
Problem 4295
| ODE |
\[ \boxed {y^{2} {y^{\prime }}^{3}+2 x y^{\prime }-y=0} \] |
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program solution |
\[ \frac {3 \ln \left (x \right )}{4} = \int _{}^{\frac {y}{x^{\frac {3}{4}}}}-\frac {9 \textit {\_a} \left (3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}+32}\right )^{\frac {1}{3}} 3^{\frac {1}{6}}}{-2 \,36^{\frac {1}{3}} {\left (\left (3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}+32}\right )^{2}\right )}^{\frac {1}{3}}+9 \,3^{\frac {1}{6}} \left (3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}+32}\right )^{\frac {1}{3}} \textit {\_a}^{2}+8 \,18^{\frac {1}{3}}}d \textit {\_a} +c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {2 \left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= \frac {2 \left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= -\frac {2 i \left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= \frac {2 i \left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \sqrt {c_{1} \left (c_{1}^{2}+2 x \right )} \\ y \left (x \right ) &= -\sqrt {c_{1} \left (c_{1}^{2}+2 x \right )} \\ \end{align*} |
Problem 4296
| ODE |
\[ \boxed {4 y^{2} {y^{\prime }}^{3}-2 x y^{\prime }+y=0} \] |
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program solution |
\[ 9 \left (\int _{}^{\frac {y}{x^{\frac {3}{4}}}}\frac {\textit {\_a} \left (3 \sqrt {3}\, \textit {\_a}^{2}-\sqrt {27 \textit {\_a}^{4}-8}\right )^{\frac {1}{3}}}{\left (2 \,3^{\frac {2}{3}}-6 i 3^{\frac {1}{6}}\right ) \left (3 \sqrt {3}\, \textit {\_a}^{2}-\sqrt {27 \textit {\_a}^{4}-8}\right )^{\frac {2}{3}}+9 \textit {\_a}^{2} \left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right ) \left (3 \sqrt {3}\, \textit {\_a}^{2}-\sqrt {27 \textit {\_a}^{4}-8}\right )^{\frac {1}{3}}-8 \,3^{\frac {2}{3}}}d \textit {\_a} \right ) \left (i 3^{\frac {2}{3}}+3^{\frac {1}{6}}\right )-c_{1} +\frac {3 \ln \left (x \right )}{4} = 0 \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{\frac {3}{4}}}{3} \\ y \left (x \right ) &= \frac {2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{\frac {3}{4}}}{3} \\ y \left (x \right ) &= -\frac {i 2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{\frac {3}{4}}}{3} \\ y \left (x \right ) &= \frac {i 2^{\frac {3}{4}} 3^{\frac {1}{4}} x^{\frac {3}{4}}}{3} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \sqrt {2}\, \sqrt {c_{1} \left (-2 c_{1}^{2}+x \right )} \\ y \left (x \right ) &= -\sqrt {2}\, \sqrt {c_{1} \left (-2 c_{1}^{2}+x \right )} \\ \end{align*} |
Problem 4297
| ODE |
\[ \boxed {16 y^{2} {y^{\prime }}^{3}+2 x y^{\prime }-y=0} \] |
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program solution |
\[ \frac {3 \ln \left (x \right )}{4} = \int _{}^{\frac {y}{x^{\frac {3}{4}}}}-\frac {9 \textit {\_a} \left (3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}+2}\right )^{\frac {1}{3}} 3^{\frac {1}{6}}}{-18^{\frac {1}{3}} {\left (\left (3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}+2}\right )^{2}\right )}^{\frac {1}{3}}+6^{\frac {2}{3}}+9 \,3^{\frac {1}{6}} \left (3 \sqrt {3}\, \textit {\_a}^{2}+\sqrt {27 \textit {\_a}^{4}+2}\right )^{\frac {1}{3}} \textit {\_a}^{2}}d \textit {\_a} +c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {\left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= \frac {\left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= -\frac {i \left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= \frac {i \left (-x^{3}\right )^{\frac {1}{4}} 6^{\frac {1}{4}}}{3} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \sqrt {2}\, \sqrt {c_{1} \left (8 c_{1}^{2}+x \right )} \\ y \left (x \right ) &= -\sqrt {2}\, \sqrt {c_{1} \left (8 c_{1}^{2}+x \right )} \\ \end{align*} |
Problem 4298
| ODE |
\[ \boxed {x y^{2} {y^{\prime }}^{3}-y^{3} {y^{\prime }}^{2}+x \left (x^{2}+1\right ) y^{\prime }-y x^{2}=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 4299
| ODE |
\[ \boxed {y^{3} {y^{\prime }}^{3}-\left (-3 x +1\right ) y^{2} {y^{\prime }}^{2}+3 x^{2} y y^{\prime }-y^{2}=-x^{3}} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {\sqrt {-6-81 x^{2}-6 \sqrt {-\left (6 x -1\right )^{3}}+54 x}}{9} \\ y \left (x \right ) &= \frac {\sqrt {-6-81 x^{2}-6 \sqrt {-\left (6 x -1\right )^{3}}+54 x}}{9} \\ y \left (x \right ) &= -\frac {\sqrt {-6-81 x^{2}+6 \sqrt {-\left (6 x -1\right )^{3}}+54 x}}{9} \\ y \left (x \right ) &= \frac {\sqrt {-6-81 x^{2}+6 \sqrt {-\left (6 x -1\right )^{3}}+54 x}}{9} \\ y \left (x \right ) &= \sqrt {-\left (c_{1}^{3}\right )^{\frac {2}{3}}+2 c_{1} x +c_{1}^{3}-x^{2}} \\ y \left (x \right ) &= -\sqrt {-\left (c_{1}^{3}\right )^{\frac {2}{3}}+2 c_{1} x +c_{1}^{3}-x^{2}} \\ y \left (x \right ) &= -\frac {\sqrt {\left (-2 i \sqrt {3}+2\right ) \left (c_{1}^{3}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, c_{1} x +4 c_{1}^{3}-4 x^{2}-4 c_{1} x}}{2} \\ y \left (x \right ) &= \frac {\sqrt {\left (-2 i \sqrt {3}+2\right ) \left (c_{1}^{3}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, c_{1} x +4 c_{1}^{3}-4 x^{2}-4 c_{1} x}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {\left (2 i \sqrt {3}+2\right ) \left (c_{1}^{3}\right )^{\frac {2}{3}}+4 i \sqrt {3}\, c_{1} x +4 c_{1}^{3}-4 x^{2}-4 c_{1} x}}{2} \\ y \left (x \right ) &= \frac {\sqrt {\left (2 i \sqrt {3}+2\right ) \left (c_{1}^{3}\right )^{\frac {2}{3}}+4 i \sqrt {3}\, c_{1} x +4 c_{1}^{3}-4 x^{2}-4 c_{1} x}}{2} \\ \end{align*} |
Problem 4300
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\[ \boxed {y^{4} {y^{\prime }}^{3}-6 x y^{\prime }+2 y=0} \] |
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program solution |
\[ \frac {\ln \left (x \right )}{2} = \int _{}^{\frac {y}{\sqrt {x}}}\frac {\textit {\_a}^{2} \left (-\textit {\_a}^{3}+\sqrt {\textit {\_a}^{6}-8}\right )^{\frac {1}{3}}}{-\left (-\textit {\_a}^{3}+\sqrt {\textit {\_a}^{6}-8}\right )^{\frac {1}{3}} \textit {\_a}^{3}+2 \left (-\textit {\_a}^{3}+\sqrt {\textit {\_a}^{6}-8}\right )^{\frac {2}{3}}+4}d \textit {\_a} +c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \sqrt {x \left (-1-i \sqrt {3}\right )} \\ y \left (x \right ) &= \sqrt {\left (-1+i \sqrt {3}\right ) x} \\ y \left (x \right ) &= -\sqrt {-\left (1+i \sqrt {3}\right ) x} \\ y \left (x \right ) &= -\sqrt {\left (-1+i \sqrt {3}\right ) x} \\ y \left (x \right ) &= \sqrt {2}\, \sqrt {x} \\ y \left (x \right ) &= -\sqrt {2}\, \sqrt {x} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} \left (-c_{1}^{3}+6 c_{1} x \right )^{\frac {1}{3}}}{2} \\ y \left (x \right ) &= -\frac {2^{\frac {2}{3}} \left (-c_{1}^{3}+6 c_{1} x \right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{4} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} \left (-c_{1}^{3}+6 c_{1} x \right )^{\frac {1}{3}} \left (-1+i \sqrt {3}\right )}{4} \\ \end{align*} |