Problem 4301
| ODE |
\[ \boxed {{y^{\prime }}^{4}-\left (y-a \right )^{3} \left (y-b \right )^{2}=0} \] |
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program solution |
\[ \int _{}^{y}\frac {1}{\left (\textit {\_a}^{5}-3 \textit {\_a}^{4} a -2 \textit {\_a}^{4} b +3 \textit {\_a}^{3} a^{2}+6 \textit {\_a}^{3} a b +\textit {\_a}^{3} b^{2}-\textit {\_a}^{2} a^{3}-6 \textit {\_a}^{2} a^{2} b -3 \textit {\_a}^{2} a \,b^{2}+2 \textit {\_a} \,a^{3} b +3 \textit {\_a} \,a^{2} b^{2}-a^{3} b^{2}\right )^{\frac {1}{4}}}d \textit {\_a} = x +c_{1} \] Verified OK. \[ \int _{}^{y}-\frac {i}{\left (\textit {\_a}^{5}-3 \textit {\_a}^{4} a -2 \textit {\_a}^{4} b +3 \textit {\_a}^{3} a^{2}+6 \textit {\_a}^{3} a b +\textit {\_a}^{3} b^{2}-\textit {\_a}^{2} a^{3}-6 \textit {\_a}^{2} a^{2} b -3 \textit {\_a}^{2} a \,b^{2}+2 \textit {\_a} \,a^{3} b +3 \textit {\_a} \,a^{2} b^{2}-a^{3} b^{2}\right )^{\frac {1}{4}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}-\frac {1}{\left (\textit {\_a}^{5}-3 \textit {\_a}^{4} a -2 \textit {\_a}^{4} b +3 \textit {\_a}^{3} a^{2}+6 \textit {\_a}^{3} a b +\textit {\_a}^{3} b^{2}-\textit {\_a}^{2} a^{3}-6 \textit {\_a}^{2} a^{2} b -3 \textit {\_a}^{2} a \,b^{2}+2 \textit {\_a} \,a^{3} b +3 \textit {\_a} \,a^{2} b^{2}-a^{3} b^{2}\right )^{\frac {1}{4}}}d \textit {\_a} = x +c_{3} \] Verified OK. \[ \int _{}^{y}\frac {i}{\left (\textit {\_a}^{5}-3 \textit {\_a}^{4} a -2 \textit {\_a}^{4} b +3 \textit {\_a}^{3} a^{2}+6 \textit {\_a}^{3} a b +\textit {\_a}^{3} b^{2}-\textit {\_a}^{2} a^{3}-6 \textit {\_a}^{2} a^{2} b -3 \textit {\_a}^{2} a \,b^{2}+2 \textit {\_a} \,a^{3} b +3 \textit {\_a} \,a^{2} b^{2}-a^{3} b^{2}\right )^{\frac {1}{4}}}d \textit {\_a} = x +c_{4} \] 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 )^{3} \left (\textit {\_a} -b \right )^{2}\right )^{\frac {1}{4}}}d \textit {\_a} \right )-c_{1} &= 0 \\ x -i \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{3} \left (\textit {\_a} -b \right )^{2}\right )^{\frac {1}{4}}}d \textit {\_a} \right )-c_{1} &= 0 \\ x +i \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{3} \left (\textit {\_a} -b \right )^{2}\right )^{\frac {1}{4}}}d \textit {\_a} \right )-c_{1} &= 0 \\ x +\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{3} \left (\textit {\_a} -b \right )^{2}\right )^{\frac {1}{4}}}d \textit {\_a} -c_{1} &= 0 \\ \end{align*} |
Problem 4302
| ODE |
\[ \boxed {{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{2}=0} \] |
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program solution |
\[
\int _{}^{y}\frac {1}{\left (\left (b -y \right )^{2} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{4}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {i}{\left (\left (b -y \right )^{2} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{4}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {1}{\left (\left (b -y \right )^{2} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{4}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}\frac {i}{\left (\left (b -y \right )^{2} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{4}}d x +c_{1}
\] Verified OK.
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Maple solution |
\begin{align*} \int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a} -a \right )^{\frac {3}{4}} \sqrt {\textit {\_a} -b}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{3} \left (y \left (x \right )-b \right )^{2}\right )^{\frac {1}{4}}d \textit {\_a}}{\left (y \left (x \right )-a \right )^{\frac {3}{4}} \sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a} -a \right )^{\frac {3}{4}} \sqrt {\textit {\_a} -b}}d \textit {\_a} +\frac {i \left (\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{3} \left (y \left (x \right )-b \right )^{2}\right )^{\frac {1}{4}}d \textit {\_a} \right )}{\left (y \left (x \right )-a \right )^{\frac {3}{4}} \sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a} -a \right )^{\frac {3}{4}} \sqrt {\textit {\_a} -b}}d \textit {\_a} -\frac {i \left (\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{3} \left (y \left (x \right )-b \right )^{2}\right )^{\frac {1}{4}}d \textit {\_a} \right )}{\left (y \left (x \right )-a \right )^{\frac {3}{4}} \sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a} -a \right )^{\frac {3}{4}} \sqrt {\textit {\_a} -b}}d \textit {\_a} +\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{3} \left (y \left (x \right )-b \right )^{2}\right )^{\frac {1}{4}}d \textit {\_a}}{\left (y \left (x \right )-a \right )^{\frac {3}{4}} \sqrt {y \left (x \right )-b}}+c_{1} &= 0 \\ \end{align*} |
Problem 4303
| ODE |
\[ \boxed {{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{3}=0} \] |
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program solution |
\[
\int _{}^{y}\frac {1}{\left (\left (b -y \right )^{3} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{4}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {i}{\left (\left (b -y \right )^{3} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{4}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {1}{\left (\left (b -y \right )^{3} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{4}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}\frac {i}{\left (\left (b -y \right )^{3} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{4}}d x +c_{1}
\] 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 {3}{4}}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{3} \left (y \left (x \right )-b \right )^{3}\right )^{\frac {1}{4}}d \textit {\_a}}{\left (\left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {3}{4}}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )\right )^{\frac {3}{4}}}d \textit {\_a} +\frac {i \left (\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{3} \left (y \left (x \right )-b \right )^{3}\right )^{\frac {1}{4}}d \textit {\_a} \right )}{\left (\left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {3}{4}}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )\right )^{\frac {3}{4}}}d \textit {\_a} -\frac {i \left (\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{3} \left (y \left (x \right )-b \right )^{3}\right )^{\frac {1}{4}}d \textit {\_a} \right )}{\left (\left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {3}{4}}}+c_{1} &= 0 \\ \int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -b \right ) \left (\textit {\_a} -a \right )\right )^{\frac {3}{4}}}d \textit {\_a} +\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{3} \left (y \left (x \right )-b \right )^{3}\right )^{\frac {1}{4}}d \textit {\_a}}{\left (\left (y \left (x \right )-b \right ) \left (y \left (x \right )-a \right )\right )^{\frac {3}{4}}}+c_{1} &= 0 \\ \end{align*} |
Problem 4304
| ODE |
\[ \boxed {{y^{\prime }}^{4}+f \left (x \right ) \left (y-a \right )^{3} \left (y-b \right )^{3} \left (y-c \right )^{2}=0} \] |
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program solution |
\[
\int _{}^{y}\frac {1}{\left (\left (c -y \right )^{2} \left (b -y \right )^{3} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{4}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {i}{\left (\left (c -y \right )^{2} \left (b -y \right )^{3} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{4}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {1}{\left (\left (c -y \right )^{2} \left (b -y \right )^{3} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{4}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}\frac {i}{\left (\left (c -y \right )^{2} \left (b -y \right )^{3} \left (a -y \right )^{3}\right )^{\frac {1}{4}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{4}}d x +c_{1}
\] Verified OK.
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Maple solution |
\[ \int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a} -a \right )^{\frac {3}{4}} \sqrt {\textit {\_a} -c}\, \left (\textit {\_a} -b \right )^{\frac {3}{4}}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-c \right )^{2} \left (y \left (x \right )-b \right )^{3} \left (y \left (x \right )-a \right )^{3}\right )^{\frac {1}{4}}d \textit {\_a}}{\left (y \left (x \right )-a \right )^{\frac {3}{4}} \sqrt {y \left (x \right )-c}\, \left (y \left (x \right )-b \right )^{\frac {3}{4}}}+c_{1} = 0 \] |
Problem 4305
| ODE |
\[ \boxed {{y^{\prime }}^{4}+x y^{\prime }-3 y=0} \] |
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program solution |
\[ y = 0 \] Verified OK. \[ x = \frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+x \textit {\_Z} -3 y\right )^{3}}{5}+c_{1} \sqrt {\operatorname {RootOf}\left (\textit {\_Z}^{4}+x \textit {\_Z} -3 y\right )} \] Verified OK.
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Maple solution |
\[ \left [x \left (\textit {\_T} \right ) = \frac {\sqrt {\textit {\_T}}\, \left (4 \textit {\_T}^{\frac {5}{2}}+5 c_{1} \right )}{5}, y \left (\textit {\_T} \right ) = \frac {3 \textit {\_T}^{4}}{5}+\frac {\textit {\_T}^{\frac {3}{2}} c_{1}}{3}\right ] \] |
Problem 4306
| ODE |
\[ \boxed {{y^{\prime }}^{4}-4 x^{2} y {y^{\prime }}^{2}+16 x y^{2} y^{\prime }-16 y^{3}=0} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {x^{4}}{16} \\ y \left (x \right ) &= 0 \\ y \left (x \right ) \left (\sqrt {x^{2}-4 \sqrt {y \left (x \right )}}-x \right )^{-\frac {2 \sqrt {x^{2} y \left (x \right )-4 y \left (x \right )^{\frac {3}{2}}}}{\sqrt {x^{2}-4 \sqrt {y \left (x \right )}}\, \sqrt {y \left (x \right )}}} \left (\sqrt {x^{2}-4 \sqrt {y \left (x \right )}}+x \right )^{\frac {2 \sqrt {x^{2} y \left (x \right )-4 y \left (x \right )^{\frac {3}{2}}}}{\sqrt {x^{2}-4 \sqrt {y \left (x \right )}}\, \sqrt {y \left (x \right )}}}-c_{1} &= 0 \\ \end{align*} |
Problem 4307
| ODE |
\[ \boxed {{y^{\prime }}^{4}+4 y {y^{\prime }}^{3}+6 y^{2} {y^{\prime }}^{2}-\left (1-4 y^{3}\right ) y^{\prime }-\left (3-y^{3}\right ) y=0} \] |
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program solution |
\[ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (\textit {\_Z}^{4}+4 \textit {\_a} \,\textit {\_Z}^{3}+6 \textit {\_Z}^{2} \textit {\_a}^{2}+\left (4 \textit {\_a}^{3}-1\right ) \textit {\_Z} +\textit {\_a}^{4}-3 \textit {\_a} \right )}d \textit {\_a} = x +c_{1} \] Verified OK.
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Maple solution | \[ x +\frac {\ln \left (\left (-14640 y \left (x \right )^{6}-93435 y \left (x \right )^{3}-256\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+4 y \left (x \right ) \textit {\_Z}^{3}+6 y \left (x \right )^{2} \textit {\_Z}^{2}+\left (4 y \left (x \right )^{3}-1\right ) \textit {\_Z} +y \left (x \right )^{4}-3 y \left (x \right )\right )^{3}+\left (-39648 y \left (x \right )^{7}-177915 y \left (x \right )^{4}+2048 y \left (x \right )\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+4 y \left (x \right ) \textit {\_Z}^{3}+6 y \left (x \right )^{2} \textit {\_Z}^{2}+\left (4 y \left (x \right )^{3}-1\right ) \textit {\_Z} +y \left (x \right )^{4}-3 y \left (x \right )\right )^{2}+\left (-36144 y \left (x \right )^{8}-162033 y \left (x \right )^{5}-9216 y \left (x \right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}+4 y \left (x \right ) \textit {\_Z}^{3}+6 y \left (x \right )^{2} \textit {\_Z}^{2}+\left (4 y \left (x \right )^{3}-1\right ) \textit {\_Z} +y \left (x \right )^{4}-3 y \left (x \right )\right )-11072 y \left (x \right )^{9}-8169 y \left (x \right )^{6}+124155 y \left (x \right )^{3}\right )}{9}-c_{1} = 0 \] |
Problem 4308
| ODE |
\[ \boxed {2 {y^{\prime }}^{4}-y^{\prime } y=2} \] |
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program solution |
\[ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (2 \textit {\_Z}^{4}-\textit {\_Z} \textit {\_a} -2\right )}d \textit {\_a} = x +c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {\sqrt {-6 \sqrt {\left (c_{1}^{2}-2 c_{1} x +x^{2}+12\right )^{3}}-6 c_{1}^{3}+18 c_{1}^{2} x +\left (-18 x^{2}+216\right ) c_{1} +6 x^{3}-216 x}}{9} \\ y \left (x \right ) &= \frac {\sqrt {-6 \sqrt {\left (c_{1}^{2}-2 c_{1} x +x^{2}+12\right )^{3}}-6 c_{1}^{3}+18 c_{1}^{2} x +\left (-18 x^{2}+216\right ) c_{1} +6 x^{3}-216 x}}{9} \\ y \left (x \right ) &= -\frac {\sqrt {6 \sqrt {\left (c_{1}^{2}-2 c_{1} x +x^{2}+12\right )^{3}}-6 c_{1}^{3}+18 c_{1}^{2} x +\left (-18 x^{2}+216\right ) c_{1} +6 x^{3}-216 x}}{9} \\ y \left (x \right ) &= \frac {\sqrt {6 \sqrt {\left (c_{1}^{2}-2 c_{1} x +x^{2}+12\right )^{3}}-6 c_{1}^{3}+18 c_{1}^{2} x +\left (-18 x^{2}+216\right ) c_{1} +6 x^{3}-216 x}}{9} \\ \end{align*} |
Problem 4309
| ODE |
\[ \boxed {x {y^{\prime }}^{4}-2 y {y^{\prime }}^{3}=-12 x^{3}} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {2 \sqrt {6}\, \left (-x \right )^{\frac {3}{2}}}{3} \\ y \left (x \right ) &= -\frac {2 \sqrt {6}\, \left (-x \right )^{\frac {3}{2}}}{3} \\ y \left (x \right ) &= -\frac {2 \sqrt {6}\, x^{\frac {3}{2}}}{3} \\ y \left (x \right ) &= \frac {2 \sqrt {6}\, x^{\frac {3}{2}}}{3} \\ y \left (x \right ) &= \frac {12 c_{1}^{4}+x^{2}}{2 c_{1}} \\ \end{align*} |
Problem 4310
| ODE |
\[ \boxed {3 {y^{\prime }}^{5}-y^{\prime } y=-1} \] |
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program solution |
\[ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (3 \textit {\_Z}^{5}-\textit {\_Z} \textit {\_a} +1\right )}d \textit {\_a} = x +c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {5 \operatorname {RootOf}\left (1+8 \textit {\_Z}^{5}+\left (2 c_{1} -2 x \right ) \textit {\_Z}^{2}\right )^{3}+2 c_{1} -2 x}{2 \operatorname {RootOf}\left (1+8 \textit {\_Z}^{5}+\left (2 c_{1} -2 x \right ) \textit {\_Z}^{2}\right ) \left (4 \operatorname {RootOf}\left (1+8 \textit {\_Z}^{5}+\left (2 c_{1} -2 x \right ) \textit {\_Z}^{2}\right )^{3}+c_{1} -x \right )} \] |
Problem 4311
| ODE |
\[ \boxed {{y^{\prime }}^{6}-\left (y-a \right )^{4} \left (y-b \right )^{3}=0} \] |
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program solution |
\[ \int _{}^{y}\frac {1}{\left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{1} \] Verified OK. \[ \int _{}^{y}\frac {2}{\left (1+i \sqrt {3}\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{2} \] Verified OK. \[ \int _{}^{y}\frac {2}{\left (i \sqrt {3}-1\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{3} \] Verified OK. \[ \int _{}^{y}-\frac {1}{\left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{4} \] Verified OK. \[ \int _{}^{y}-\frac {2}{\left (1+i \sqrt {3}\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{5} \] Verified OK. \[ \int _{}^{y}-\frac {2}{\left (i \sqrt {3}-1\right ) \left (\textit {\_a}^{7}-4 \textit {\_a}^{6} a -3 \textit {\_a}^{6} b +6 \textit {\_a}^{5} a^{2}+12 \textit {\_a}^{5} a b +3 \textit {\_a}^{5} b^{2}-4 \textit {\_a}^{4} a^{3}-18 \textit {\_a}^{4} a^{2} b -12 \textit {\_a}^{4} a \,b^{2}-\textit {\_a}^{4} b^{3}+\textit {\_a}^{3} a^{4}+12 \textit {\_a}^{3} a^{3} b +18 \textit {\_a}^{3} a^{2} b^{2}+4 \textit {\_a}^{3} a \,b^{3}-3 \textit {\_a}^{2} a^{4} b -12 \textit {\_a}^{2} a^{3} b^{2}-6 \textit {\_a}^{2} a^{2} b^{3}+3 \textit {\_a} \,a^{4} b^{2}+4 \textit {\_a} \,a^{3} b^{3}-a^{4} b^{3}\right )^{\frac {1}{6}}}d \textit {\_a} = x +c_{6} \] 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 )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} \right )-c_{1} &= 0 \\ \frac {2 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}-c_{1} +x}{1+i \sqrt {3}} &= 0 \\ \frac {-2 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+c_{1} -x}{-1+i \sqrt {3}} &= 0 \\ \frac {2 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+c_{1} -x}{-1+i \sqrt {3}} &= 0 \\ \frac {-2 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}-c_{1} +x}{1+i \sqrt {3}} &= 0 \\ x +\int _{}^{y \left (x \right )}\frac {1}{\left (\left (\textit {\_a} -a \right )^{4} \left (\textit {\_a} -b \right )^{3}\right )^{\frac {1}{6}}}d \textit {\_a} -c_{1} &= 0 \\ \end{align*} |
Problem 4312
| ODE |
\[ \boxed {{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{4} \left (y-b \right )^{3}=0} \] |
|
program solution |
\[
\int _{}^{y}\frac {1}{\left (\left (b -y \right )^{3} \left (a -y \right )^{4}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}\frac {2}{\left (1+i \sqrt {3}\right ) \left (\left (b -y \right )^{3} \left (a -y \right )^{4}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}\frac {2}{\left (i \sqrt {3}-1\right ) \left (\left (b -y \right )^{3} \left (a -y \right )^{4}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {1}{\left (\left (b -y \right )^{3} \left (a -y \right )^{4}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {2}{\left (1+i \sqrt {3}\right ) \left (\left (b -y \right )^{3} \left (a -y \right )^{4}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {2}{\left (i \sqrt {3}-1\right ) \left (\left (b -y \right )^{3} \left (a -y \right )^{4}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK.
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Maple solution |
\[ \int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a} -a \right )^{\frac {2}{3}} \sqrt {\textit {\_a} -b}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-a \right )^{4} \left (y \left (x \right )-b \right )^{3}\right )^{\frac {1}{6}}d \textit {\_a}}{\left (y \left (x \right )-a \right )^{\frac {2}{3}} \sqrt {y \left (x \right )-b}}+c_{1} = 0 \] |
Problem 4313
| ODE |
\[ \boxed {{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{5} \left (y-b \right )^{3}=0} \] |
|
program solution |
\[
\int _{}^{y}\frac {1}{\left (\left (b -y \right )^{3} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}\frac {2}{\left (1+i \sqrt {3}\right ) \left (\left (b -y \right )^{3} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}\frac {2}{\left (i \sqrt {3}-1\right ) \left (\left (b -y \right )^{3} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {1}{\left (\left (b -y \right )^{3} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {2}{\left (1+i \sqrt {3}\right ) \left (\left (b -y \right )^{3} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {2}{\left (i \sqrt {3}-1\right ) \left (\left (b -y \right )^{3} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int \left (-f \left (x \right )\right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK.
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Maple solution |
\[ \int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a} -a \right )^{\frac {5}{6}} \sqrt {\textit {\_a} -b}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-b \right )^{3} \left (y \left (x \right )-a \right )^{5}\right )^{\frac {1}{6}}d \textit {\_a}}{\left (y \left (x \right )-a \right )^{\frac {5}{6}} \sqrt {y \left (x \right )-b}}+c_{1} = 0 \] |
Problem 4314
| ODE |
\[ \boxed {{y^{\prime }}^{6}+f \left (x \right ) \left (y-a \right )^{5} \left (y-b \right )^{4}=0} \] |
|
program solution |
\[
\int _{}^{y}\frac {1}{\left (\left (b -y \right )^{4} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}\frac {2}{\left (1+i \sqrt {3}\right ) \left (\left (b -y \right )^{4} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}\frac {2}{\left (i \sqrt {3}-1\right ) \left (\left (b -y \right )^{4} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {1}{\left (\left (b -y \right )^{4} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {2}{\left (1+i \sqrt {3}\right ) \left (\left (b -y \right )^{4} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK. \[
\int _{}^{y}-\frac {2}{\left (i \sqrt {3}-1\right ) \left (\left (b -y \right )^{4} \left (a -y \right )^{5}\right )^{\frac {1}{6}}}d \textit {\_a} = \int f \left (x \right )^{\frac {1}{6}}d x +c_{1}
\] Verified OK.
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Maple solution |
\[ \int _{}^{y \left (x \right )}\frac {1}{\left (\textit {\_a} -a \right )^{\frac {5}{6}} \left (\textit {\_a} -b \right )^{\frac {2}{3}}}d \textit {\_a} -\frac {\int _{}^{x}\left (-f \left (\textit {\_a} \right ) \left (y \left (x \right )-b \right )^{4} \left (y \left (x \right )-a \right )^{5}\right )^{\frac {1}{6}}d \textit {\_a}}{\left (y \left (x \right )-a \right )^{\frac {5}{6}} \left (y \left (x \right )-b \right )^{\frac {2}{3}}}+c_{1} = 0 \] |
Problem 4315
| ODE |
\[ \boxed {x^{2} \left ({y^{\prime }}^{6}+3 y^{4}+3 y^{2}+1\right )=a^{2}} \] |
|
program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 4316
| ODE |
\[ \boxed {2 \sqrt {a y^{\prime }}+x y^{\prime }-y=0} \] |
|
program solution |
\[ y = c_{1} x +2 \sqrt {c_{1} a} \] Verified OK. \[ y = \frac {2 \sqrt {\frac {a^{2}}{x^{2}}}\, x +a}{x} \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = 2 \sqrt {c_{1} a}+c_{1} x \] |
Problem 4317
| ODE |
\[ \boxed {\left (-y+x \right ) \sqrt {y^{\prime }}-a \left (y^{\prime }+1\right )=0} \] |
|
program solution |
\[ y = x -2 a \] Verified OK. \[ x = \frac {a \sqrt {2}}{\sqrt {\frac {x \sqrt {y^{2}-2 y x -4 a^{2}+x^{2}}-y \sqrt {y^{2}-2 y x -4 a^{2}+x^{2}}-2 a^{2}+x^{2}-2 y x +y^{2}}{a^{2}}}}+c_{2} \] Verified OK. \[ x = \frac {a \sqrt {2}}{\sqrt {\frac {-x \sqrt {y^{2}-2 y x -4 a^{2}+x^{2}}+y \sqrt {y^{2}-2 y x -4 a^{2}+x^{2}}-2 a^{2}+x^{2}-2 y x +y^{2}}{a^{2}}}}+c_{2} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= x -2 a \\ y \left (x \right ) &= x -\frac {a \left (a^{2}+\left (x -c_{1} \right )^{2}\right )}{\sqrt {\frac {a^{2}}{\left (c_{1} -x \right )^{2}}}\, \left (x -c_{1} \right )^{2}} \\ \end{align*} |
Problem 4318
| ODE |
\[ \boxed {2 \left (1+y\right )^{\frac {3}{2}}+3 x y^{\prime }-3 y=0} \] |
|
program solution |
\[ -\ln \left (x \right )-\frac {3 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 \textit {\_Z}^{3}+3 \textit {\_Z}^{2}+12 \textit {\_Z} +4\right )}{\sum }\frac {\textit {\_R} \ln \left (y-\textit {\_R} \right )}{2 \textit {\_R}^{2}+\textit {\_R} +2}\right )}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{3}-3 \textit {\_Z}^{2}+3\right )}{\sum }\frac {\textit {\_R} \left (2 \textit {\_R} -3\right ) \ln \left (\sqrt {1+y}-\textit {\_R} \right )}{\textit {\_R}^{2}-\textit {\_R}}\right )}{6}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{3}+3 \textit {\_Z}^{2}-3\right )}{\sum }\frac {\textit {\_R} \left (2 \textit {\_R} +3\right ) \ln \left (\sqrt {1+y}-\textit {\_R} \right )}{\textit {\_R}^{2}+\textit {\_R}}\right )}{6}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{3}-3 \textit {\_Z}^{2}+3\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\sqrt {1+y}-\textit {\_R} \right )}{\textit {\_R}^{2}-\textit {\_R}}\right )}{3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{3}+3 \textit {\_Z}^{2}-3\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\sqrt {1+y}-\textit {\_R} \right )}{\textit {\_R}^{2}+\textit {\_R}}\right )}{3} = c_{1} \] Verified OK.
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Maple solution | \[ \ln \left (x \right )+3 \left (\int _{}^{y \left (x \right )}\frac {1}{\left (2 \textit {\_a} +2\right ) \sqrt {\textit {\_a} +1}-3 \textit {\_a}}d \textit {\_a} \right )+c_{1} = 0 \] |
Problem 4319
| ODE |
\[ \boxed {\sqrt {1+{y^{\prime }}^{2}}+a y^{\prime }=x} \] |
|
program solution |
\[ y = \frac {\frac {x \sqrt {a^{2}+x^{2}-1}}{2}+\frac {\left (4 a^{2}-4\right ) \ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )}{8}+\frac {a \,x^{2}}{2}}{a^{2}-1}+c_{1} \] Verified OK. \[ y = -\frac {\frac {x \sqrt {a^{2}+x^{2}-1}}{2}+\frac {\left (4 a^{2}-4\right ) \ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )}{8}-\frac {a \,x^{2}}{2}}{a^{2}-1}+c_{2} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {a \,x^{2}+x \sqrt {a^{2}+x^{2}-1}+\left (a -1\right ) \left (\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )+2 c_{1} \right ) \left (a +1\right )}{2 a^{2}-2} \\ y \left (x \right ) &= \frac {a \,x^{2}-x \sqrt {a^{2}+x^{2}-1}-\left (a -1\right ) \left (a +1\right ) \left (\ln \left (x +\sqrt {a^{2}+x^{2}-1}\right )-2 c_{1} \right )}{2 a^{2}-2} \\ \end{align*} |
Problem 4320
| ODE |
\[ \boxed {\sqrt {1+{y^{\prime }}^{2}}+a y^{\prime }-y=0} \] |
|
program solution |
\[ \int _{}^{y}\frac {a^{2}-1}{a \textit {\_a} +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} = x +c_{1} \] Verified OK. \[ \int _{}^{y}\frac {a^{2}-1}{a \textit {\_a} -\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} = x +c_{2} \] Verified OK.
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Maple solution |
\begin{align*} -\left (\int _{}^{y \left (x \right )}\frac {1}{\textit {\_a} a +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} \right ) a^{2}+\int _{}^{y \left (x \right )}\frac {1}{\textit {\_a} a +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} -c_{1} +x &= 0 \\ \left (\int _{}^{y \left (x \right )}\frac {1}{-\textit {\_a} a +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} \right ) a^{2}-\left (\int _{}^{y \left (x \right )}\frac {1}{-\textit {\_a} a +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} \right )-c_{1} +x &= 0 \\ \end{align*} |
Problem 4321
| ODE |
\[ \boxed {\sqrt {1+{y^{\prime }}^{2}}-x y^{\prime }=0} \] |
|
program solution |
\[ y = \ln \left (x +\sqrt {x^{2}-1}\right )+c_{1} \] Verified OK. \[ y = \ln \left (x -\sqrt {x^{2}-1}\right )+c_{2} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \ln \left (x +\sqrt {x^{2}-1}\right )+c_{1} \\ y \left (x \right ) &= -\ln \left (x +\sqrt {x^{2}-1}\right )+c_{1} \\ \end{align*} |
Problem 4322
| ODE |
\[ \boxed {\sqrt {a^{2}+b^{2} {y^{\prime }}^{2}}+x y^{\prime }-y=0} \] |
|
program solution |
\[ y = c_{1} x +\sqrt {b^{2} c_{1}^{2}+a^{2}} \] Verified OK. \[ y = \frac {\sqrt {\frac {a^{2} b^{2}}{b^{2}-x^{2}}}\, \sqrt {b^{2}-x^{2}}\, b +a \,x^{2}}{\sqrt {b^{2}-x^{2}}\, b} \] Verified OK. \[ y = -\frac {a \,x^{2}-\sqrt {\frac {a^{2} b^{2}}{b^{2}-x^{2}}}\, \sqrt {b^{2}-x^{2}}\, b}{\sqrt {b^{2}-x^{2}}\, b} \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = \sqrt {b^{2} c_{1}^{2}+a^{2}}+c_{1} x \] |
Problem 4323
| ODE |
\[ \boxed {a \sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y=0} \] |
|
program solution |
\[ y = c_{1} x +a \sqrt {c_{1}^{2}+1} \] Verified OK. \[ y = \frac {a \sqrt {\frac {a^{2}}{a^{2}-x^{2}}}\, \sqrt {a^{2}-x^{2}}+x^{2}}{\sqrt {a^{2}-x^{2}}} \] Verified OK. \[ y = \frac {a \sqrt {\frac {a^{2}}{a^{2}-x^{2}}}\, \sqrt {a^{2}-x^{2}}-x^{2}}{\sqrt {a^{2}-x^{2}}} \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = a \sqrt {c_{1}^{2}+1}+c_{1} x \] |
Problem 4324
| ODE |
\[ \boxed {a x \sqrt {1+{y^{\prime }}^{2}}+x y^{\prime }-y=0} \] |
|
program solution |
\[ y = -i x \] Verified OK. \[ y = i x \] Verified OK.
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|
Maple solution |
\begin{align*} \frac {x \sqrt {\frac {-a^{2} x^{2}+y \left (x \right )^{2} a^{2}+2 \sqrt {y \left (x \right )^{2}-a^{2} x^{2}+x^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-{\mathrm e}^{\frac {\operatorname {arcsinh}\left (\frac {\sqrt {y \left (x \right )^{2}-a^{2} x^{2}+x^{2}}\, a +y \left (x \right )}{\left (a^{2}-1\right ) x}\right )}{a}} c_{1}}{\sqrt {\frac {-a^{2} x^{2}+y \left (x \right )^{2} a^{2}+2 \sqrt {y \left (x \right )^{2}-a^{2} x^{2}+x^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} &= 0 \\ \frac {x \sqrt {\frac {-a^{2} x^{2}+y \left (x \right )^{2} a^{2}-2 \sqrt {y \left (x \right )^{2}-a^{2} x^{2}+x^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}-{\mathrm e}^{\frac {\operatorname {arcsinh}\left (\frac {-\sqrt {y \left (x \right )^{2}-a^{2} x^{2}+x^{2}}\, a +y \left (x \right )}{\left (a^{2}-1\right ) x}\right )}{a}} c_{1}}{\sqrt {\frac {-a^{2} x^{2}+y \left (x \right )^{2} a^{2}-2 \sqrt {y \left (x \right )^{2}-a^{2} x^{2}+x^{2}}\, a y \left (x \right )+x^{2}+y \left (x \right )^{2}}{\left (a^{2}-1\right )^{2} x^{2}}}} &= 0 \\ \end{align*} |
Problem 4325
| ODE |
\[ \boxed {\sqrt {\left (a \,x^{2}+y^{2}\right ) \left (1+{y^{\prime }}^{2}\right )}-y^{\prime } y=a x} \] |
|
program solution |
\[ y = -\left (-a \right )^{\frac {3}{2}} x +\sqrt {-a^{3}+2 a^{2}-a}\, x \] Verified OK. \[ y = \left (-a \right )^{\frac {3}{2}} x +\sqrt {-a^{3}+2 a^{2}-a}\, x \] Verified OK. \[ x = c_{2} {\left (\frac {\sqrt {\left (a -1\right ) a}\, a y+\sqrt {\frac {\left (a^{2} x^{2}+2 y^{2} a +2 \sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, y-y^{2}\right ) \left (a -1\right )}{x^{2}}}\, a x +\sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, \sqrt {\left (a -1\right ) a}}{a x}\right )}^{-\frac {a}{\sqrt {\left (a -1\right ) a}}} \] Warning, solution could not be verified \[ x = c_{2} {\left (\frac {\sqrt {\frac {\left (a^{2} x^{2}+2 y^{2} a -2 \sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, y-y^{2}\right ) \left (a -1\right )}{x^{2}}}\, a x +\sqrt {\left (a -1\right ) a}\, \left (a y-\sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\right )}{a x}\right )}^{-\frac {a}{\sqrt {\left (a -1\right ) a}}} \] Warning, solution could not be verified \[ y = -\left (-a \right )^{\frac {3}{2}} x -\sqrt {-a^{3}+2 a^{2}-a}\, x \] Verified OK. \[ y = \left (-a \right )^{\frac {3}{2}} x -\sqrt {-a^{3}+2 a^{2}-a}\, x \] Verified OK. \[ x = c_{4} {\left (\frac {\sqrt {\left (a -1\right ) a}\, a y+\sqrt {\frac {\left (a^{2} x^{2}+2 y^{2} a +2 \sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, y-y^{2}\right ) \left (a -1\right )}{x^{2}}}\, a x +\sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, \sqrt {\left (a -1\right ) a}}{a x}\right )}^{\frac {a}{\sqrt {\left (a -1\right ) a}}} \] Warning, solution could not be verified \[ x = c_{4} {\left (\frac {\sqrt {\frac {\left (a^{2} x^{2}+2 y^{2} a -2 \sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\, y-y^{2}\right ) \left (a -1\right )}{x^{2}}}\, a x +\sqrt {\left (a -1\right ) a}\, \left (a y-\sqrt {a \left (a -1\right ) \left (a \,x^{2}+y^{2}\right )}\right )}{a x}\right )}^{\frac {a}{\sqrt {\left (a -1\right ) a}}} \] Warning, solution could not be verified |
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Maple solution |
\begin{align*} y \left (x \right ) &= \sqrt {-a}\, x \\ y \left (x \right ) &= -\sqrt {-a}\, x \\ y \left (x \right ) &= -\frac {a^{2} \left (a -1\right ) x^{\frac {-\sqrt {a \left (a -1\right )}+a}{a}}-x^{\frac {a +\sqrt {a \left (a -1\right )}}{a}} c_{1}^{2}}{2 \sqrt {a \left (a -1\right )}\, c_{1}} \\ y \left (x \right ) &= -\frac {-x^{\frac {-\sqrt {a \left (a -1\right )}+a}{a}} c_{1}^{2}+x^{\frac {a +\sqrt {a \left (a -1\right )}}{a}} a^{2} \left (a -1\right )}{2 \sqrt {a \left (a -1\right )}\, c_{1}} \\ \end{align*} |
Problem 4326
| ODE |
\[ \boxed {a \left (1+{y^{\prime }}^{3}\right )^{\frac {1}{3}}+x y^{\prime }-y=0} \] |
|
program solution |
\[ y = c_{1} x +a \left (c_{1}^{3}+1\right )^{\frac {1}{3}} \] Verified OK. \[ y = \frac {{\left (x \left (\sqrt {-a x}\, a -x^{2}\right ) \left (a^{3}+x^{3}\right )^{2}\right )}^{\frac {1}{3}} x +\left (a^{4}+a \,x^{3}\right ) {\left (\frac {a \left (x \sqrt {-a x}+a^{2}\right )}{a^{3}+x^{3}}\right )}^{\frac {1}{3}}}{a^{3}+x^{3}} \] Verified OK. \[ y = \frac {\left (i \sqrt {3}-1\right ) x {\left (x \left (\sqrt {-a x}\, a -x^{2}\right ) \left (a^{3}+x^{3}\right )^{2}\right )}^{\frac {1}{3}}+\left (2 a^{4}+2 a \,x^{3}\right ) {\left (\frac {a \left (x \sqrt {-a x}+a^{2}\right )}{a^{3}+x^{3}}\right )}^{\frac {1}{3}}}{2 a^{3}+2 x^{3}} \] Verified OK. \[ y = \frac {\left (2 a^{4}+2 a \,x^{3}\right ) {\left (\frac {a \left (x \sqrt {-a x}+a^{2}\right )}{a^{3}+x^{3}}\right )}^{\frac {1}{3}}-\left (1+i \sqrt {3}\right ) x {\left (x \left (\sqrt {-a x}\, a -x^{2}\right ) \left (a^{3}+x^{3}\right )^{2}\right )}^{\frac {1}{3}}}{2 a^{3}+2 x^{3}} \] Verified OK. \[ y = \frac {{\left (-x \left (\sqrt {-a x}\, a +x^{2}\right ) \left (a^{3}+x^{3}\right )^{2}\right )}^{\frac {1}{3}} x +\left (a^{4}+a \,x^{3}\right ) {\left (\frac {a \left (a^{2}-x \sqrt {-a x}\right )}{a^{3}+x^{3}}\right )}^{\frac {1}{3}}}{a^{3}+x^{3}} \] Verified OK. \[ y = \munderset {p \rightarrow \frac {\left (i \sqrt {3}-1\right ) {\left (-x \left (\sqrt {-a x}\, a +x^{2}\right ) \left (a^{3}+x^{3}\right )^{2}\right )}^{\frac {1}{3}}}{2 a^{3}+2 x^{3}}}{\operatorname {lim}}\left (a \left (p^{3}+1\right )^{\frac {1}{3}}+x p \right ) \] Verified OK. \[ y = \munderset {p \rightarrow -\frac {\left (1+i \sqrt {3}\right ) {\left (-x \left (\sqrt {-a x}\, a +x^{2}\right ) \left (a^{3}+x^{3}\right )^{2}\right )}^{\frac {1}{3}}}{2 a^{3}+2 x^{3}}}{\operatorname {lim}}\left (a \left (p^{3}+1\right )^{\frac {1}{3}}+x p \right ) \] Verified OK.
|
|
Maple solution |
\[ y \left (x \right ) = a \left (c_{1}^{3}+1\right )^{\frac {1}{3}}+c_{1} x \] |
Problem 4327
| ODE |
\[ \boxed {\cos \left (y^{\prime }\right )+x y^{\prime }-y=0} \] |
|
program solution |
\[ y = c_{1} x +\cos \left (c_{1} \right ) \] Verified OK. \[ y = x \arcsin \left (x \right )+\sqrt {-x^{2}+1} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \arcsin \left (x \right ) x +\sqrt {-x^{2}+1} \\ y \left (x \right ) &= \cos \left (c_{1} \right )+c_{1} x \\ \end{align*} |
Problem 4328
| ODE |
\[ \boxed {a \cos \left (y^{\prime }\right )+b y^{\prime }=-x} \] |
|
program solution |
\[ y = \int \operatorname {RootOf}\left (a \cos \left (\textit {\_Z} \right )+\textit {\_Z} b +x \right )d x +c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \int \operatorname {RootOf}\left (a \cos \left (\textit {\_Z} \right )+\textit {\_Z} b +x \right )d x +c_{1} \] |
Problem 4329
| ODE |
\[ \boxed {\sin \left (y^{\prime }\right )+y^{\prime }=x} \] |
|
program solution |
\[ y = \int \operatorname {RootOf}\left (\sin \left (\textit {\_Z} \right )+\textit {\_Z} -x \right )d x +c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \int \operatorname {RootOf}\left (\sin \left (\textit {\_Z} \right )+\textit {\_Z} -x \right )d x +c_{1} \] |
Problem 4330
| ODE |
\[ \boxed {y^{\prime } \sin \left (y^{\prime }\right )+\cos \left (y^{\prime }\right )-y=0} \] |
|
program solution |
\[ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (\textit {\_Z} \sin \left (\textit {\_Z} \right )+\cos \left (\textit {\_Z} \right )-\textit {\_a} \right )}d \textit {\_a} = x +c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= 1 \\ x -\left (\int _{}^{y \left (x \right )}\frac {1}{\operatorname {RootOf}\left (\textit {\_Z} \sin \left (\textit {\_Z} \right )+\cos \left (\textit {\_Z} \right )-\textit {\_a} \right )}d \textit {\_a} \right )-c_{1} &= 0 \\ \end{align*} |
Problem 4331
| ODE |
\[ \boxed {{y^{\prime }}^{2} \left (x +\sin \left (y^{\prime }\right )\right )-y=0} \] |
|
program solution |
\[ y = 0 \] Verified OK. \[ y = \sin \left (1\right )+x \] Verified OK. \[ x = \frac {\left (-\operatorname {RootOf}\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z}^{2}+x \,\textit {\_Z}^{2}-y\right )^{2}+\operatorname {RootOf}\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z}^{2}+x \,\textit {\_Z}^{2}-y\right )\right ) \sin \left (\operatorname {RootOf}\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z}^{2}+x \,\textit {\_Z}^{2}-y\right )\right )+c_{1} -\cos \left (\operatorname {RootOf}\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z}^{2}+x \,\textit {\_Z}^{2}-y\right )\right )}{{\left (\operatorname {RootOf}\left (\sin \left (\textit {\_Z} \right ) \textit {\_Z}^{2}+x \,\textit {\_Z}^{2}-y\right )-1\right )}^{2}} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= 0 \\ \left [x \left (\textit {\_T} \right ) &= \frac {\left (-\textit {\_T}^{2}+\textit {\_T} \right ) \sin \left (\textit {\_T} \right )-\cos \left (\textit {\_T} \right )+c_{1}}{\left (\textit {\_T} -1\right )^{2}}, y \left (\textit {\_T} \right ) &= -\frac {\left (\left (\textit {\_T} -1\right ) \sin \left (\textit {\_T} \right )+\cos \left (\textit {\_T} \right )-c_{1} \right ) \textit {\_T}^{2}}{\left (\textit {\_T} -1\right )^{2}}\right ] \\ \end{align*} |
Problem 4332
| ODE |
\[ \boxed {\left (1+{y^{\prime }}^{2}\right ) \sin \left (x y^{\prime }-y\right )^{2}=1} \] |
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program solution |
\[ y = c_{1} x -\arcsin \left (\frac {1}{\sqrt {c_{1}^{2}+1}}\right ) \] Verified OK. \[ y = -\sqrt {\frac {1}{x}}\, \sqrt {1-x}\, x -\arcsin \left (\frac {1}{\sqrt {\frac {1}{x}}}\right ) \] Verified OK. \[ y = \sqrt {x +1}\, \sqrt {-\frac {1}{x}}\, x -\arcsin \left (\frac {1}{\sqrt {-\frac {1}{x}}}\right ) \] Verified OK. \[ y = c_{2} x +\arcsin \left (\frac {1}{\sqrt {c_{2}^{2}+1}}\right ) \] Verified OK. \[ y = \sqrt {\frac {1}{x}}\, \sqrt {1-x}\, x +\arcsin \left (\frac {1}{\sqrt {\frac {1}{x}}}\right ) \] Verified OK. \[ y = -\sqrt {x +1}\, \sqrt {-\frac {1}{x}}\, x +\arcsin \left (\frac {1}{\sqrt {-\frac {1}{x}}}\right ) \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -x \sqrt {\frac {1}{x}}\, \sqrt {1-x}-\arcsin \left (\frac {1}{\sqrt {\frac {1}{x}}}\right ) \\ y \left (x \right ) &= x \sqrt {\frac {1}{x}}\, \sqrt {1-x}+\arcsin \left (\frac {1}{\sqrt {\frac {1}{x}}}\right ) \\ y \left (x \right ) &= -x \sqrt {-\frac {1}{x}}\, \sqrt {x +1}+\arcsin \left (\frac {1}{\sqrt {-\frac {1}{x}}}\right ) \\ y \left (x \right ) &= x \sqrt {-\frac {1}{x}}\, \sqrt {x +1}-\arcsin \left (\frac {1}{\sqrt {-\frac {1}{x}}}\right ) \\ y \left (x \right ) &= c_{1} x -\arcsin \left (\frac {1}{\sqrt {c_{1}^{2}+1}}\right ) \\ y \left (x \right ) &= c_{1} x +\arcsin \left (\frac {1}{\sqrt {c_{1}^{2}+1}}\right ) \\ \end{align*} |
Problem 4333
| ODE |
\[ \boxed {\left (1+{y^{\prime }}^{2}\right ) \left (\arctan \left (y^{\prime }\right )+a x \right )+y^{\prime }=0} \] |
|
program solution |
\[ y = \int \tan \left (\operatorname {RootOf}\left (a x \tan \left (\textit {\_Z} \right )^{2}+\tan \left (\textit {\_Z} \right )^{2} \textit {\_Z} +a x +\tan \left (\textit {\_Z} \right )+\textit {\_Z} \right )\right )d x +c_{1} \] Warning, solution could not be verified |
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Maple solution | \[ y \left (x \right ) = \int \tan \left (\operatorname {RootOf}\left (a x +\sin \left (\textit {\_Z} \right ) \cos \left (\textit {\_Z} \right )+\textit {\_Z} \right )\right )d x +c_{1} \] |
Problem 4334
| ODE |
\[ \boxed {-{y^{\prime }}^{2}=-{\mathrm e}^{y^{\prime }-y}-1} \] |
|
program solution |
\[ \int _{}^{y}\frac {1}{\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z} -\textit {\_a}}+\textit {\_Z}^{2}-1\right )}d \textit {\_a} = x +c_{1} \] Verified OK.
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Maple solution | \[ x -\left (\int _{}^{y \left (x \right )}\frac {1}{\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z} -\textit {\_a}}+\textit {\_Z}^{2}-1\right )}d \textit {\_a} \right )-c_{1} = 0 \] |
Problem 4335
| ODE |
\[ \boxed {\ln \left (y^{\prime }\right )+x y^{\prime }=-a} \] |
|
program solution |
\[ y = \frac {\operatorname {LambertW}\left (x \,{\mathrm e}^{-a}\right )^{2}}{2}+\operatorname {LambertW}\left (x \,{\mathrm e}^{-a}\right )+c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\operatorname {LambertW}\left (x \,{\mathrm e}^{-a}\right )^{2}}{2}+\operatorname {LambertW}\left (x \,{\mathrm e}^{-a}\right )+c_{1} \] |
Problem 4336
| ODE |
\[ \boxed {\ln \left (y^{\prime }\right )+x y^{\prime }-y=-a} \] |
|
program solution |
\[ y = c_{1} x +\ln \left (c_{1} \right )+a \] Verified OK. \[ y = \ln \left (-\frac {1}{x}\right )+a -1 \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \ln \left (-\frac {1}{x}\right )+a -1 \\ y \left (x \right ) &= \ln \left (c_{1} \right )+c_{1} x +a \\ \end{align*} |
Problem 4337
| ODE |
\[ \boxed {\ln \left (y^{\prime }\right )+x y^{\prime }+b y=-a} \] |
|
program solution |
\[ y = \frac {\infty }{\operatorname {signum}\left (b \right )} \] Warning, solution could not be verified \[ x = \frac {c_{1} {\left (\frac {\operatorname {LambertW}\left (x \,{\mathrm e}^{-b y-a}\right )}{x}\right )}^{-\frac {1}{b +1}} \operatorname {LambertW}\left (x \,{\mathrm e}^{-b y-a}\right ) b +x}{\operatorname {LambertW}\left (x \,{\mathrm e}^{-b y-a}\right ) b} \] Verified OK.
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Maple solution |
\[ \frac {-\left ({\left (\frac {\operatorname {LambertW}\left (x \,{\mathrm e}^{-b y \left (x \right )-a}\right )}{x}\right )}^{-\frac {1}{b +1}} c_{1} -x \right ) b \operatorname {LambertW}\left (x \,{\mathrm e}^{-b y \left (x \right )-a}\right )-x}{b \operatorname {LambertW}\left (x \,{\mathrm e}^{-b y \left (x \right )-a}\right )} = 0 \] |
Problem 4338
| ODE |
\[ \boxed {\ln \left (y^{\prime }\right )+4 x y^{\prime }-2 y=0} \] |
|
program solution |
\[ y = -\infty \] Warning, solution could not be verified \[ x = -\frac {2 x \left (-8 c_{1} x +\operatorname {LambertW}\left (4 x \,{\mathrm e}^{2 y}\right )\right )}{\operatorname {LambertW}\left (4 x \,{\mathrm e}^{2 y}\right )^{2}} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -\ln \left (2\right )+\frac {\ln \left (\frac {-1+\sqrt {16 c_{1} x +1}}{x}\right )}{2}-\frac {1}{2}+\frac {\sqrt {16 c_{1} x +1}}{2} \\ y \left (x \right ) &= -\ln \left (2\right )+\frac {\ln \left (\frac {-1-\sqrt {16 c_{1} x +1}}{x}\right )}{2}-\frac {1}{2}-\frac {\sqrt {16 c_{1} x +1}}{2} \\ \end{align*} |
Problem 4339
| ODE |
\[ \boxed {\ln \left (y^{\prime }\right )+a \left (x y^{\prime }-y\right )=0} \] |
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program solution |
\[ y = c_{1} x +\frac {\ln \left (c_{1} \right )}{a} \] Verified OK. \[ y = \frac {\ln \left (-\frac {1}{a x}\right )-1}{a} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {\ln \left (-\frac {1}{a x}\right )-1}{a} \\ y \left (x \right ) &= c_{1} x +\frac {\ln \left (c_{1} \right )}{a} \\ \end{align*} |
Problem 4340
| ODE |
\[ \boxed {a \left (\ln \left (y^{\prime }\right )-y^{\prime }\right )+y=x} \] |
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program solution |
\[ y = a +x \] Verified OK. \[ y = x +a \left ({\mathrm e}^{\frac {x}{a}} c_{1} -\ln \left ({\mathrm e}^{\frac {x}{a}} c_{1} \right )\right ) \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= a +x \\ y \left (x \right ) &= -a \ln \left ({\mathrm e}^{\frac {x -c_{1}}{a}}\right )+a \,{\mathrm e}^{\frac {x -c_{1}}{a}}+x \\ \end{align*} |
Problem 4341
| ODE |
\[ \boxed {y \ln \left (y^{\prime }\right )+y^{\prime }-y \ln \left (y\right )-y x=0} \] |
|
program solution |
\[ y = x c_{2} {\mathrm e}^{\frac {\operatorname {LambertW}\left ({\mathrm e}^{x}\right )^{2}}{2}+\operatorname {LambertW}\left ({\mathrm e}^{x}\right )-\ln \left (x \right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {\operatorname {LambertW}\left ({\mathrm e}^{x}\right ) \left (\operatorname {LambertW}\left ({\mathrm e}^{x}\right )+2\right )}{2}} \] |
Problem 4342
| ODE |
\[ \boxed {y^{\prime } \ln \left (y^{\prime }\right )-\left (x +1\right ) y^{\prime }+y=0} \] |
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program solution |
\[ y = c_{1} x -c_{1} \ln \left (c_{1} \right )+c_{1} \] Verified OK. \[ y = {\mathrm e}^{x} \left (1-\ln \left ({\mathrm e}^{x}\right )+x \right ) \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= {\mathrm e}^{x} \\ y \left (x \right ) &= c_{1} \left (-\ln \left (c_{1} \right )+x +1\right ) \\ \end{align*} |
Problem 4343
| ODE |
\[ \boxed {y^{\prime } \ln \left (y^{\prime }+\sqrt {a +{y^{\prime }}^{2}}\right )-\sqrt {1+{y^{\prime }}^{2}}-x y^{\prime }+y=0} \] |
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program solution |
\[ y = c_{1} x -c_{1} \ln \left (c_{1} +\sqrt {c_{1}^{2}+a}\right )+\sqrt {c_{1}^{2}+1} \] Verified OK. \[ \text {Expression too large to display} \] Warning, solution could not be verified |
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Maple solution | \[ \text {No solution found} \] |
Problem 4344
| ODE |
\[ \boxed {\ln \left (\cos \left (y^{\prime }\right )\right )+y^{\prime } \tan \left (y^{\prime }\right )-y=0} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= 0 \\ x -\left (\int _{}^{y \left (x \right )}\frac {1}{\operatorname {RootOf}\left (\ln \left (\cos \left (\textit {\_Z} \right )\right )+\textit {\_Z} \tan \left (\textit {\_Z} \right )-\textit {\_a} \right )}d \textit {\_a} \right )-c_{1} &= 0 \\ \end{align*} |
Problem 4345
| ODE |
\[ \boxed {y^{\prime }-\frac {x y}{x^{2}-y^{2}}=0} \] |
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program solution |
\[ y = {\mathrm e}^{\frac {\operatorname {LambertW}\left (-x^{2} {\mathrm e}^{-2 c_{1}}\right )}{2}+c_{1}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \sqrt {-\frac {1}{\operatorname {LambertW}\left (-c_{1} x^{2}\right )}}\, x \] |
Problem 4346
| ODE |
\[ \boxed {y^{\prime }-\frac {x +y-3}{x -y-1}=0} \] |
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program solution |
\[ \frac {\ln \left (y^{2}+x^{2}-2 y-4 x +5\right )}{2}-\arctan \left (\frac {y-1}{x -2}\right ) = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 1+\tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x -2\right )+2 c_{1} \right )\right ) \left (-x +2\right ) \] |
Problem 4347
| ODE |
\[ \boxed {y^{\prime }-\frac {2 x +y-1}{4 x +2 y+5}=0} \] |
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program solution |
\[ y = \frac {{\mathrm e}^{-\operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {25 x}{7}+\frac {18}{7}-\frac {25 c_{1}}{7}}}{7}\right )+\frac {25 x}{7}+\frac {18}{7}-\frac {25 c_{1}}{7}}}{5}-2 x -\frac {9}{5} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {7 \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{\frac {18}{7}+\frac {25 x}{7}-\frac {25 c_{1}}{7}}}{7}\right )}{10}-\frac {9}{5}-2 x \] |
Problem 4348
| ODE |
\[ \boxed {y^{\prime }-\frac {2 y}{x +1}=\left (x +1\right )^{2}} \] |
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program solution |
\[ y = \left (x +1\right )^{2} \left (x +c_{1} \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (x +c_{1} \right ) \left (1+x \right )^{2} \] |
Problem 4349
| ODE |
\[ \boxed {y^{\prime }+y x -x^{3} y^{3}=0} \] |
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program solution |
\[ y = \frac {1}{\sqrt {x^{2}+1+c_{1} {\mathrm e}^{x^{2}}}} \] Verified OK. \[ y = -\frac {1}{\sqrt {x^{2}+1+c_{1} {\mathrm e}^{x^{2}}}} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= \frac {1}{\sqrt {{\mathrm e}^{x^{2}} c_{1} +x^{2}+1}} \\ y \left (x \right ) &= -\frac {1}{\sqrt {{\mathrm e}^{x^{2}} c_{1} +x^{2}+1}} \\ \end{align*} |
Problem 4350
| ODE |
\[ \boxed {\frac {2 x}{y^{3}}+\frac {\left (y^{2}-3 x^{2}\right ) y^{\prime }}{y^{4}}=0} \] |
|
program solution |
\[ \frac {x^{2}}{y^{3}}-\frac {1}{y} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {1+\frac {\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}}}{3 c_{1}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {2}{3}}-4 i \sqrt {3}-4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}+4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {2}{3}}-4 i \sqrt {3}+4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}-4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}} c_{1}} \\ \end{align*} |
Problem 4351
| ODE |
\[ \boxed {y+x y^{2}-x y^{\prime }=0} \] |
|
program solution |
\[ y = -\frac {2 x}{x^{2}+c_{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {2 x}{x^{2}-2 c_{1}} \] |
Problem 4352
| ODE |
\[ \boxed {y^{2} \left (1+{y^{\prime }}^{2}\right )=R^{2}} \] |
|
program solution |
\[ -\sqrt {-y^{2}+R^{2}} = x +c_{1} \] Verified OK. \[ \sqrt {-y^{2}+R^{2}} = x +c_{2} \] Verified OK.
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Maple solution |
\begin{align*} y \left (x \right ) &= -R \\ y \left (x \right ) &= R \\ y \left (x \right ) &= \sqrt {R^{2}-c_{1}^{2}+2 c_{1} x -x^{2}} \\ y \left (x \right ) &= -\sqrt {\left (R +c_{1} -x \right ) \left (R -c_{1} +x \right )} \\ \end{align*} |
Problem 4353
| ODE |
\[ \boxed {y-x y^{\prime }-\frac {a y^{\prime }}{\sqrt {1+{y^{\prime }}^{2}}}=0} \] |
|
program solution |
\[ y = c_{1} x +\frac {a c_{1}}{\sqrt {c_{1}^{2}+1}} \] Verified OK. \[ y = \frac {\sqrt {\left (-a \,x^{2}\right )^{\frac {2}{3}}-x^{2}}\, \left (x \sqrt {\frac {\left (-a \,x^{2}\right )^{\frac {2}{3}}}{x^{2}}}+a \right )}{x \sqrt {\frac {\left (-a \,x^{2}\right )^{\frac {2}{3}}}{x^{2}}}} \] Verified OK. \[ y = \frac {\left (-x \sqrt {\frac {\left (-a \,x^{2}\right )^{\frac {2}{3}}}{x^{2}}}-a \right ) \sqrt {\left (-a \,x^{2}\right )^{\frac {2}{3}}-x^{2}}}{x \sqrt {\frac {\left (-a \,x^{2}\right )^{\frac {2}{3}}}{x^{2}}}} \] Verified OK. \[ y = \frac {\sqrt {i \sqrt {3}\, \left (-a \,x^{2}\right )^{\frac {2}{3}}-\left (-a \,x^{2}\right )^{\frac {2}{3}}-2 x^{2}}\, \left (\sqrt {2}\, x \sqrt {\frac {\left (i \sqrt {3}-1\right ) \left (-a \,x^{2}\right )^{\frac {2}{3}}}{x^{2}}}+2 a \right )}{2 x \sqrt {\frac {\left (i \sqrt {3}-1\right ) \left (-a \,x^{2}\right )^{\frac {2}{3}}}{x^{2}}}} \] Verified OK. \[ y = -\frac {\sqrt {i \sqrt {3}\, \left (-a \,x^{2}\right )^{\frac {2}{3}}-\left (-a \,x^{2}\right )^{\frac {2}{3}}-2 x^{2}}\, \left (\frac {\sqrt {2}\, x \sqrt {\frac {\left (i \sqrt {3}-1\right ) \left (-a \,x^{2}\right )^{\frac {2}{3}}}{x^{2}}}}{2}+a \right )}{\sqrt {\frac {\left (i \sqrt {3}-1\right ) \left (-a \,x^{2}\right )^{\frac {2}{3}}}{x^{2}}}\, x} \] Verified OK. \[ y = \frac {\sqrt {-i \sqrt {3}\, \left (-a \,x^{2}\right )^{\frac {2}{3}}-\left (-a \,x^{2}\right )^{\frac {2}{3}}-2 x^{2}}\, \left (\sqrt {2}\, x \sqrt {-\frac {\left (1+i \sqrt {3}\right ) \left (-a \,x^{2}\right )^{\frac {2}{3}}}{x^{2}}}+2 a \right )}{2 x \sqrt {-\frac {\left (1+i \sqrt {3}\right ) \left (-a \,x^{2}\right )^{\frac {2}{3}}}{x^{2}}}} \] Verified OK. \[ y = -\frac {\sqrt {-i \sqrt {3}\, \left (-a \,x^{2}\right )^{\frac {2}{3}}-\left (-a \,x^{2}\right )^{\frac {2}{3}}-2 x^{2}}\, \left (\frac {\sqrt {2}\, x \sqrt {-\frac {\left (1+i \sqrt {3}\right ) \left (-a \,x^{2}\right )^{\frac {2}{3}}}{x^{2}}}}{2}+a \right )}{\sqrt {-\frac {\left (1+i \sqrt {3}\right ) \left (-a \,x^{2}\right )^{\frac {2}{3}}}{x^{2}}}\, x} \] Verified OK.
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Maple solution |
\[ y \left (x \right ) = c_{1} \left (x +\frac {a}{\sqrt {c_{1}^{2}+1}}\right ) \] |
Problem 4354
| ODE |
\[ \boxed {y-x {y^{\prime }}^{2}-{y^{\prime }}^{2}=0} \] |
|
program solution |
\[ y = 0 \] Verified OK. \[ y = x +1 \] Verified OK. \[ y = x \left (1+\frac {c_{1}}{\sqrt {x +1}}\right )^{2}+\left (1+\frac {c_{1}}{\sqrt {x +1}}\right )^{2} \] Verified OK.
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|
Maple solution |
\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {\left (x +1+\sqrt {\left (1+x \right ) \left (1+c_{1} \right )}\right )^{2}}{1+x} \\ y \left (x \right ) &= \frac {\left (-x -1+\sqrt {\left (1+x \right ) \left (1+c_{1} \right )}\right )^{2}}{1+x} \\ \end{align*} |
Problem 4355
| ODE |
\[ \boxed {\left (x +1\right ) y+\left (1-y\right ) x y^{\prime }=0} \] |
|
program solution |
\[ y = -\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-x -c_{1}}}{x}\right ) \] Verified OK.
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|
Maple solution | \[ y \left (x \right ) = -\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-x}}{c_{1} x}\right ) \] |
Problem 4356
| ODE |
\[ \boxed {y^{2}+x y^{2}+\left (x^{2}-y x^{2}\right ) y^{\prime }=0} \] |
|
program solution |
\[ y = {\mathrm e}^{\frac {\ln \left (x \right ) x +\operatorname {LambertW}\left (-{\mathrm e}^{-\frac {\ln \left (x \right ) x +c_{1} x -1}{x}}\right ) x +c_{1} x -1}{x}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x \,{\mathrm e}^{\frac {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {-c_{1} x +1}{x}}}{x}\right ) x +c_{1} x -1}{x}} \] |
Problem 4357
| ODE |
\[ \boxed {x y \left (x^{2}+1\right ) y^{\prime }-y^{2}=1} \] |
|
program solution |
\[ -\ln \left (x \right )+\frac {\ln \left (x^{2}+1\right )}{2}+\frac {\ln \left (1+y^{2}\right )}{2} = c_{1} \] Verified OK.
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|
Maple solution | \begin{align*} y \left (x \right ) &= \frac {\sqrt {\left (x^{2}+1\right ) \left (c_{1} x^{2}-1\right )}}{x^{2}+1} \\ y \left (x \right ) &= -\frac {\sqrt {\left (x^{2}+1\right ) \left (c_{1} x^{2}-1\right )}}{x^{2}+1} \\ \end{align*} |
Problem 4358
| ODE |
\[ \boxed {y^{2}-\left (y+\sqrt {1+y^{2}}\right ) \left (x^{2}+1\right )^{\frac {3}{2}} y^{\prime }=-1} \] |
|
program solution |
\[ -\frac {x}{\sqrt {x^{2}+1}}+\frac {\ln \left (1+y^{2}\right )}{2}+\operatorname {arcsinh}\left (y\right ) = c_{1} \] Verified OK.
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Maple solution | \[ \frac {x}{\sqrt {x^{2}+1}}-\operatorname {arcsinh}\left (y \left (x \right )\right )-\frac {\ln \left (1+y \left (x \right )^{2}\right )}{2}+c_{1} = 0 \] |
Problem 4359
| ODE |
\[ \boxed {\sin \left (x \right ) \cos \left (y\right )-\cos \left (x \right ) \sin \left (y\right ) y^{\prime }=0} \] |
|
program solution |
\[ \ln \left (\cos \left (x \right )\right )-\ln \left (\cos \left (y\right )\right ) = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \arccos \left (\frac {\cos \left (x \right )}{c_{1}}\right ) \] |
Problem 4360
| ODE |
\[ \boxed {\sec \left (x \right )^{2} \tan \left (y\right )+\sec \left (y\right )^{2} \tan \left (x \right ) y^{\prime }=0} \] |
|
program solution |
\[ -\ln \left (\tan \left (x \right )\right )-\ln \left (\tan \left (y\right )\right ) = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\arctan \left (\frac {2 \tan \left (x \right ) c_{1}}{c_{1}^{2} \tan \left (x \right )^{2}+1}, \frac {c_{1}^{2} \tan \left (x \right )^{2}-1}{c_{1}^{2} \tan \left (x \right )^{2}+1}\right )}{2} \] |
Problem 4361
| ODE |
\[ \boxed {\left (y-x \right ) y^{\prime }+y=0} \] |
|
program solution |
\[ y = {\mathrm e}^{\operatorname {LambertW}\left (-x \,{\mathrm e}^{-c_{1}}\right )+c_{1}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {x}{\operatorname {LambertW}\left (-x \,{\mathrm e}^{-c_{1}}\right )} \] |
Problem 4362
| ODE |
\[ \boxed {\left (2 \sqrt {y x}-x \right ) y^{\prime }+y=0} \] |
|
program solution |
\[ \frac {\ln \left (y\right ) \sqrt {y}+\sqrt {x}}{\sqrt {y}} = c_{1} \] Verified OK.
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Maple solution | \[ \ln \left (y \left (x \right )\right )+\frac {x}{\sqrt {x y \left (x \right )}}-c_{1} = 0 \] |
Problem 4363
| ODE |
\[ \boxed {x y^{\prime }-y-\sqrt {x^{2}+y^{2}}=0} \] |
|
program solution |
\[ y = -\frac {{\mathrm e}^{-c_{1}} \left ({\mathrm e}^{2 c_{1}}-x^{2}\right )}{2} \] Verified OK.
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Maple solution | \[ \frac {-c_{1} x^{2}+\sqrt {x^{2}+y \left (x \right )^{2}}+y \left (x \right )}{x^{2}} = 0 \] |
Problem 4364
| ODE |
\[ \boxed {-y \cos \left (\frac {y}{x}\right )+x \cos \left (\frac {y}{x}\right ) y^{\prime }=-x} \] |
|
program solution |
\[ \sin \left (\frac {y}{x}\right )-\ln \left (\frac {1}{x}\right ) = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\arcsin \left (\ln \left (x \right )+c_{1} \right ) x \] |
Problem 4365
| ODE |
\[ \boxed {8 y+\left (5 y+7 x \right ) y^{\prime }=-10 x} \] |
|
program solution |
\[ \frac {3 \ln \left (2 x +y\right )}{5}+\frac {2 \ln \left (y+x \right )}{5} = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x \left (\operatorname {RootOf}\left (\textit {\_Z}^{25} c_{1} x^{5}-2 \textit {\_Z}^{20} c_{1} x^{5}+\textit {\_Z}^{15} c_{1} x^{5}-1\right )^{5}-2\right ) \] |
Problem 4366
| ODE |
\[ \boxed {-y+\left (2 y-1\right ) y^{\prime }=-1-2 x} \] |
|
program solution |
\[ \frac {\ln \left (8 x^{2}+\left (-4 y+6\right ) x +8 y^{2}-9 y+3\right )}{4}-\frac {\sqrt {15}\, \arctan \left (\frac {\left (-16 y+4 x +9\right ) \sqrt {15}}{60 x +15}\right )}{30} = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\sqrt {15}\, \tan \left (\operatorname {RootOf}\left (\sqrt {15}\, \ln \left (\left (1+4 x \right )^{2} \sec \left (\textit {\_Z} \right )^{2}\right )-3 \sqrt {15}\, \ln \left (2\right )+\sqrt {15}\, \ln \left (3\right )+\sqrt {15}\, \ln \left (5\right )+2 \sqrt {15}\, c_{1} -2 \textit {\_Z} \right )\right ) \left (-1-4 x \right )}{16}+\frac {x}{4}+\frac {9}{16} \] |
Problem 4367
| ODE |
\[ \boxed {3 y+\left (7 y-3 x +3\right ) y^{\prime }=7 x -7} \] |
|
program solution |
\[ \frac {2 \ln \left (-x +y+1\right )}{3}+\frac {5 \ln \left (x -1+y\right )}{3} = c_{1} \] Verified OK.
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Maple solution | \[ \text {Expression too large to display} \] |
Problem 4368
| ODE |
\[ \boxed {y^{\prime }+\frac {x y}{x^{2}+1}=\frac {1}{2 x \left (x^{2}+1\right )}} \] |
|
program solution |
\[ y = -\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )-2 c_{1}}{2 \sqrt {x^{2}+1}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )+2 c_{1}}{2 \sqrt {x^{2}+1}} \] |
Problem 4369
| ODE |
\[ \boxed {x \left (-x^{2}+1\right ) y^{\prime }+\left (2 x^{2}-1\right ) y=a \,x^{3}} \] |
|
program solution |
\[ y = -c_{1} \sqrt {x^{2}-1}\, x +a x \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x \left (\sqrt {x -1}\, \sqrt {1+x}\, c_{1} +a \right ) \] |
Problem 4370
| ODE |
\[ \boxed {y^{\prime }+\frac {y}{\left (-x^{2}+1\right )^{\frac {3}{2}}}=\frac {x +\sqrt {-x^{2}+1}}{\left (-x^{2}+1\right )^{2}}} \] |
|
program solution |
\[ \int _{}^{x}-\frac {\left (\sqrt {-\textit {\_a}^{2}+1}\, \textit {\_a} +\left (y-1\right ) \textit {\_a}^{2}-y+1\right ) {\mathrm e}^{\frac {\textit {\_a}}{\sqrt {-\textit {\_a}^{2}+1}}}}{\left (-\textit {\_a}^{2}+1\right )^{\frac {5}{2}}}d \textit {\_a} +\left (-\left (\int _{}^{x}\frac {{\mathrm e}^{\frac {\textit {\_a}}{\sqrt {-\textit {\_a}^{2}+1}}}}{\left (-\textit {\_a}^{2}+1\right )^{\frac {3}{2}}}d \textit {\_a} \right )+{\mathrm e}^{\frac {x}{\sqrt {-x^{2}+1}}}\right ) y = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (\int \frac {{\mathrm e}^{\frac {x}{\sqrt {-x^{2}+1}}} \left (x +\sqrt {-x^{2}+1}\right )}{\left (x -1\right )^{2} \left (1+x \right )^{2}}d x +c_{1} \right ) {\mathrm e}^{-\frac {x}{\sqrt {-x^{2}+1}}} \] |
Problem 4371
| ODE |
\[ \boxed {y^{\prime }+y \cos \left (x \right )=\frac {\sin \left (2 x \right )}{2}} \] |
|
program solution |
\[ y = {\mathrm e}^{-\sin \left (x \right )} \left (\sin \left (x \right ) {\mathrm e}^{\sin \left (x \right )}-{\mathrm e}^{\sin \left (x \right )}+c_{1} \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \sin \left (x \right )-1+{\mathrm e}^{-\sin \left (x \right )} c_{1} \] |
Problem 4372
| ODE |
\[ \boxed {\left (x^{2}+1\right ) y^{\prime }+y=\arctan \left (x \right )} \] |
|
program solution |
\[ \left (y-\arctan \left (x \right )+1\right ) {\mathrm e}^{\arctan \left (x \right )} = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \arctan \left (x \right )-1+{\mathrm e}^{-\arctan \left (x \right )} c_{1} \] |
Problem 4373
| ODE |
\[ \boxed {\left (-x^{2}+1\right ) z^{\prime }-x z-a x z^{2}=0} \] |
|
program solution |
\[ z = -\frac {1}{a \left (c_{3} \sqrt {x^{2}-1}+1\right )} \] Verified OK.
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Maple solution | \[ z \left (x \right ) = \frac {1}{\sqrt {x -1}\, \sqrt {1+x}\, c_{1} -a} \] |
Problem 4374
| ODE |
\[ \boxed {3 z^{2} z^{\prime }-a z^{3}=x +1} \] |
|
program solution |
\[ \frac {\left (z^{3} a^{2}+a x +a +1\right ) {\mathrm e}^{-a x}}{a^{2}} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} z \left (x \right ) &= \frac {{\left (\left ({\mathrm e}^{a x} c_{1} a^{2}-1+\left (-1-x \right ) a \right ) a \right )}^{\frac {1}{3}}}{a} \\ z \left (x \right ) &= -\frac {{\left (\left ({\mathrm e}^{a x} c_{1} a^{2}-1+\left (-1-x \right ) a \right ) a \right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2 a} \\ z \left (x \right ) &= \frac {{\left (\left ({\mathrm e}^{a x} c_{1} a^{2}-1+\left (-1-x \right ) a \right ) a \right )}^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{2 a} \\ \end{align*} |
Problem 4375
| ODE |
\[ \boxed {z^{\prime }+2 x z-2 a \,x^{3} z^{3}=0} \] |
|
program solution |
\[ z = \frac {2}{\sqrt {4 a \,x^{2}+4 c_{1} {\mathrm e}^{2 x^{2}}+2 a}} \] Verified OK. \[ z = -\frac {2}{\sqrt {4 a \,x^{2}+4 c_{1} {\mathrm e}^{2 x^{2}}+2 a}} \] Verified OK.
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Maple solution |
\begin{align*} z \left (x \right ) &= -\frac {2}{\sqrt {4 a \,x^{2}+4 \,{\mathrm e}^{2 x^{2}} c_{1} +2 a}} \\ z \left (x \right ) &= \frac {2}{\sqrt {4 a \,x^{2}+4 \,{\mathrm e}^{2 x^{2}} c_{1} +2 a}} \\ \end{align*} |
Problem 4376
| ODE |
\[ \boxed {z^{\prime }+z \cos \left (x \right )-z^{n} \sin \left (2 x \right )=0} \] |
|
program solution |
\[ z^{-n +1} = \frac {\left (n -1\right ) c_{1} {\mathrm e}^{\left (n -1\right ) \sin \left (x \right )}+2+\left (2 n -2\right ) \sin \left (x \right )}{n -1} \] Verified OK.
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Maple solution | \[ z \left (x \right ) = \left (\frac {{\mathrm e}^{\sin \left (x \right ) \left (n -1\right )} c_{1} n -{\mathrm e}^{\sin \left (x \right ) \left (n -1\right )} c_{1} +2 \sin \left (x \right ) n -2 \sin \left (x \right )+2}{n -1}\right )^{-\frac {1}{n -1}} \] |
Problem 4377
| ODE |
\[ \boxed {x y^{\prime }+y-\ln \left (x \right ) y^{2}=0} \] |
|
program solution |
\[ y = \frac {1}{-c_{3} x +\ln \left (x \right )+1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {1}{1+c_{1} x +\ln \left (x \right )} \] |
Problem 4378
| ODE |
\[ \boxed {3 x y^{2}+\left (y^{3}+3 y x^{2}\right ) y^{\prime }=-x^{3}} \] |
|
program solution |
\[ \frac {\left (3 y^{2}+x^{2}\right )^{2}}{4}-2 y^{4} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {\sqrt {-3 c_{1} x^{2}-\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\ y \left (x \right ) &= \frac {\sqrt {-3 c_{1} x^{2}+\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\ y \left (x \right ) &= -\frac {\sqrt {-3 c_{1} x^{2}-\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\ y \left (x \right ) &= -\frac {\sqrt {-3 c_{1} x^{2}+\sqrt {8 c_{1}^{2} x^{4}+1}}}{\sqrt {c_{1}}} \\ \end{align*} |
Problem 4379
| ODE |
\[ \boxed {\frac {y^{2}}{x^{2}}-\frac {2 y y^{\prime }}{x}=-1} \] |
|
program solution |
\[ x -\frac {y^{2}}{x} = c_{1} \] 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} \\ \end{align*} |
Problem 4380
| ODE |
\[ \boxed {\frac {3 x}{y^{3}}+\left (\frac {1}{y^{2}}-\frac {3 x^{2}}{y^{4}}\right ) y^{\prime }=0} \] |
|
program solution |
\[ y = {\mathrm e}^{\frac {\operatorname {LambertW}\left (-3 x^{2} {\mathrm e}^{-2 c_{1}}\right )}{2}+c_{1}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \sqrt {3}\, \sqrt {-\frac {1}{\operatorname {LambertW}\left (-3 c_{1} x^{2}\right )}}\, x \] |
Problem 4381
| ODE |
\[ \boxed {y^{\prime } y+\frac {x y^{\prime }}{x^{2}+y^{2}}-\frac {y}{x^{2}+y^{2}}=-x} \] |
|
program solution |
\[ \frac {x^{2}}{2}-\arctan \left (\frac {x}{y}\right )+\frac {y^{2}}{2} = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \cot \left (\operatorname {RootOf}\left (2 c_{1} \sin \left (\textit {\_Z} \right )^{2}-2 \textit {\_Z} \sin \left (\textit {\_Z} \right )^{2}+x^{2}\right )\right ) x \] |
Problem 4382
| ODE |
\[ \boxed {{\mathrm e}^{\frac {x}{y}}+{\mathrm e}^{\frac {x}{y}} \left (1-\frac {x}{y}\right ) y^{\prime }=-1} \] |
|
program solution |
\[ y = -\frac {x}{\operatorname {LambertW}\left (-\frac {x}{c_{1} -x}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {x}{\operatorname {LambertW}\left (\frac {x c_{1}}{c_{1} x -1}\right )} \] |
Problem 4383
| ODE |
\[ \boxed {{\mathrm e}^{x} \left (x^{2}+y^{2}+2 x \right )+2 y \,{\mathrm e}^{x} y^{\prime }=0} \] |
|
program solution |
\[ \left (x^{2}+y^{2}\right ) {\mathrm e}^{x} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \sqrt {{\mathrm e}^{-x} c_{1} -x^{2}} \\ y \left (x \right ) &= -\sqrt {{\mathrm e}^{-x} c_{1} -x^{2}} \\ \end{align*} |
Problem 4384
| ODE |
\[ \boxed {n \cos \left (x n +m y\right )-m \sin \left (m x +n y\right )+\left (m \cos \left (x n +m y\right )-n \sin \left (m x +n y\right )\right ) y^{\prime }=0} \] |
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program solution |
\[ \cos \left (m x +n y\right )+\sin \left (x n +m y\right ) = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-n x +\operatorname {RootOf}\left (2 m^{2} x -2 n^{2} x -2 \arcsin \left (\sin \left (\textit {\_Z} \right )+c_{1} \right ) m -m \pi +2 \textit {\_Z} n \right )}{m} \] |
Problem 4385
| ODE |
\[ \boxed {\frac {x}{\sqrt {1+x^{2}+y^{2}}}+\frac {y y^{\prime }}{\sqrt {1+x^{2}+y^{2}}}+\frac {y}{x^{2}+y^{2}}-\frac {x y^{\prime }}{x^{2}+y^{2}}=0} \] |
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program solution |
\[ \arctan \left (\frac {x}{y}\right )+\sqrt {1+x^{2}+y^{2}} = c_{1} \] Verified OK.
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Maple solution | \[ \arctan \left (\frac {x}{y \left (x \right )}\right )+\sqrt {1+x^{2}+y \left (x \right )^{2}}-c_{1} = 0 \] |
Problem 4386
| ODE |
\[ \boxed {\frac {x^{n} y^{\prime }}{b y^{2}-c \,x^{2 a}}-\frac {a y x^{a -1}}{b y^{2}-c \,x^{2 a}}=-x^{a -1}} \] |
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program solution |
\[ y = \frac {x^{-a +n +1} \left (\frac {\partial }{\partial x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )+\frac {\textit {\_Y}^{\prime }\left (x \right ) \left (1-a +n -a \,x^{a -n}\right )}{x}-b c \,x^{4 a -2 n -2} \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right )}{b \operatorname {DESol}\left (\left \{\frac {-b c \textit {\_Y} \left (x \right ) x^{4 a -2 n}-a \,x^{a -n +1} \textit {\_Y}^{\prime }\left (x \right )-\left (-\textit {\_Y}^{\prime \prime }\left (x \right ) x +\textit {\_Y}^{\prime }\left (x \right ) \left (a -1-n \right )\right ) x}{x^{2}}\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
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Maple solution | \[ \text {No solution found} \] |
Problem 4387
| ODE |
\[ \boxed {2 y x +\left (y^{2}-2 x^{2}\right ) y^{\prime }=0} \] |
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program solution |
\[ y = {\mathrm e}^{\frac {\operatorname {LambertW}\left (-2 x^{2} {\mathrm e}^{-2 c_{1}}\right )}{2}+c_{1}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \sqrt {2}\, \sqrt {-\frac {1}{\operatorname {LambertW}\left (-2 c_{1} x^{2}\right )}}\, x \] |
Problem 4388
| ODE |
\[ \boxed {\frac {2}{y}-\frac {2 y^{\prime }}{x}=-\frac {1}{x}-\frac {y^{\prime }}{y}} \] |
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program solution |
\[ -x \left (y+x \right )+y^{2} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {c_{1} x -\sqrt {5 x^{2} c_{1}^{2}+4}}{2 c_{1}} \\ y \left (x \right ) &= \frac {c_{1} x +\sqrt {5 x^{2} c_{1}^{2}+4}}{2 c_{1}} \\ \end{align*} |
Problem 4389
| ODE |
\[ \boxed {x y^{\prime }-y-\sqrt {x^{2}+y^{2}}=0} \] |
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program solution |
\[ y = -\frac {{\mathrm e}^{-c_{1}} \left ({\mathrm e}^{2 c_{1}}-x^{2}\right )}{2} \] Verified OK.
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Maple solution | \[ \frac {-c_{1} x^{2}+\sqrt {x^{2}+y \left (x \right )^{2}}+y \left (x \right )}{x^{2}} = 0 \] |
Problem 4390
| ODE |
\[ \boxed {8 y+\left (5 y+7 x \right ) y^{\prime }=-10 x} \] |
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program solution |
\[ \frac {3 \ln \left (2 x +y\right )}{5}+\frac {2 \ln \left (y+x \right )}{5} = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x \left (\operatorname {RootOf}\left (\textit {\_Z}^{25} c_{1} x^{5}-2 \textit {\_Z}^{20} c_{1} x^{5}+\textit {\_Z}^{15} c_{1} x^{5}-1\right )^{5}-2\right ) \] |
Problem 4391
| ODE |
\[ \boxed {2 y x -y^{2}+\left (y^{2}+2 y x -x^{2}\right ) y^{\prime }=-x^{2}} \] |
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program solution |
\[ \ln \left (x^{2}+y^{2}\right )-\ln \left (y+x \right ) = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {1-\sqrt {-4 x^{2} c_{1}^{2}+4 c_{1} x +1}}{2 c_{1}} \\ y \left (x \right ) &= \frac {1+\sqrt {-4 x^{2} c_{1}^{2}+4 c_{1} x +1}}{2 c_{1}} \\ \end{align*} |
Problem 4392
| ODE |
\[ \boxed {y^{2}+\left (y x +x^{2}\right ) y^{\prime }=0} \] |
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program solution |
\[ -\frac {\ln \left (x +2 y\right )}{2}+\ln \left (y\right ) = -\frac {\ln \left (x \right )}{2}+c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {1+\sqrt {c_{1} x^{2}+1}}{c_{1} x} \\ y \left (x \right ) &= \frac {1-\sqrt {c_{1} x^{2}+1}}{c_{1} x} \\ \end{align*} |
Problem 4393
| ODE |
\[ \boxed {\left (x \cos \left (\frac {y}{x}\right )+y \sin \left (\frac {y}{x}\right )\right ) y+\left (x \cos \left (\frac {y}{x}\right )-y \sin \left (\frac {y}{x}\right )\right ) x y^{\prime }=0} \] |
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program solution |
\[ \cos \left (\frac {y}{x}\right ) y x = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x \operatorname {RootOf}\left (\textit {\_Z} \cos \left (\textit {\_Z} \right ) x^{2}-c_{1} \right ) \] |
Problem 4394
| ODE |
\[ \boxed {\left (y^{2} x^{2}+y x \right ) y+\left (y^{2} x^{2}-1\right ) x y^{\prime }=0} \] |
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program solution |
\[ y = {\mathrm e}^{-\operatorname {LambertW}\left (-x \,{\mathrm e}^{-c_{1}}\right )-c_{1}} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {1}{x} \\ y \left (x \right ) &= -\frac {\operatorname {LambertW}\left (-x \,{\mathrm e}^{-c_{1}}\right )}{x} \\ \end{align*} |
Problem 4395
| ODE |
\[ \boxed {\left (x^{3} y^{3}+y^{2} x^{2}+y x +1\right ) y+\left (x^{3} y^{3}-y^{2} x^{2}-y x +1\right ) x y^{\prime }=0} \] |
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program solution |
\[ \frac {y^{2} x^{2}-1}{x y}-2 \ln \left (y\right ) = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -\frac {1}{x} \\ y \left (x \right ) &= \frac {{\mathrm e}^{\operatorname {RootOf}\left (-{\mathrm e}^{2 \textit {\_Z}}-2 \ln \left (x \right ) {\mathrm e}^{\textit {\_Z}}+2 c_{1} {\mathrm e}^{\textit {\_Z}}+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+1\right )}}{x} \\ \end{align*} |
Problem 4396
| ODE |
\[ \boxed {y^{2}+2 y^{\prime } y=-x^{2}-2 x} \] |
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program solution |
\[ \left (x^{2}+y^{2}\right ) {\mathrm e}^{x} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \sqrt {{\mathrm e}^{-x} c_{1} -x^{2}} \\ y \left (x \right ) &= -\sqrt {{\mathrm e}^{-x} c_{1} -x^{2}} \\ \end{align*} |
Problem 4397
| ODE |
\[ \boxed {y^{2}-2 x y y^{\prime }=-x^{2}} \] |
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program solution |
\[ x -\frac {y^{2}}{x} = c_{1} \] 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} \\ \end{align*} |
Problem 4398
| ODE |
\[ \boxed {2 y x +\left (y^{2}-3 x^{2}\right ) y^{\prime }=0} \] |
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program solution |
\[ \frac {x^{2}}{y^{3}}-\frac {1}{y} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {1+\frac {\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}}{2}+\frac {2}{\left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}}}{3 c_{1}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {2}{3}}-4 i \sqrt {3}-4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}+4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {2}{3}}-4 i \sqrt {3}+4 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}}-4}{12 \left (12 \sqrt {3}\, x \sqrt {27 x^{2} c_{1}^{2}-4}\, c_{1} -108 x^{2} c_{1}^{2}+8\right )^{\frac {1}{3}} c_{1}} \\ \end{align*} |
Problem 4399
| ODE |
\[ \boxed {y+\left (-x +2 y\right ) y^{\prime }=0} \] |
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program solution |
\[ y = {\mathrm e}^{\operatorname {LambertW}\left (-\frac {x \,{\mathrm e}^{-\frac {c_{1}}{2}}}{2}\right )+\frac {c_{1}}{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {x}{2 \operatorname {LambertW}\left (-\frac {x \,{\mathrm e}^{-\frac {c_{1}}{2}}}{2}\right )} \] |
Problem 4400
| ODE |
\[ \boxed {x y^{\prime }-a y+y^{2}=x^{-2 a}} \] |
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program solution |
\[ y = \frac {\left (\left (a \,x^{a}-c_{3} \right ) \cosh \left (\frac {x^{-a}}{a}\right )+\sinh \left (\frac {x^{-a}}{a}\right ) \left (a c_{3} x^{a}-1\right )\right ) x^{-a}}{c_{3} \sinh \left (\frac {x^{-a}}{a}\right )+\cosh \left (\frac {x^{-a}}{a}\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (-x^{-a} c_{1} +a \right ) \sinh \left (\frac {x^{-a}}{a}\right )+\left (c_{1} a -x^{-a}\right ) \cosh \left (\frac {x^{-a}}{a}\right )}{\cosh \left (\frac {x^{-a}}{a}\right ) c_{1} +\sinh \left (\frac {x^{-a}}{a}\right )} \] |