Problem 1001
| ODE |
\[ \boxed {y^{\prime } x y+y^{2}=-x^{2}} \] With initial conditions \begin {align*} [y \left (1\right ) = 2] \end {align*}
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program solution |
\[ \frac {x^{4}}{4}+\frac {x^{2} y^{2}}{2} = {\frac {9}{4}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\sqrt {-2 x^{4}+18}}{2 x} \] |
Problem 1002
| ODE |
\[ \boxed {y^{\prime }-\frac {y^{2}-3 y x -5 x^{2}}{x^{2}}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
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program solution |
\[ y = -\frac {x \left (2 x^{6}-5\right )}{2 x^{6}+1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-2 x^{7}+5 x}{2 x^{6}+1} \] |
Problem 1003
| ODE |
\[ \boxed {y^{\prime } x^{2}-y^{2}-4 y x=2 x^{2}} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
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program solution |
\[ y = -\frac {x \left (4 x -3\right )}{-3+2 x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-4 x^{2}+3 x}{2 x -3} \] |
Problem 1004
| ODE |
\[ \boxed {y^{\prime } x y-4 y^{2}=3 x^{2}} \] With initial conditions \begin {align*} \left [y \left (1\right ) = \sqrt {3}\right ] \end {align*}
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program solution |
\[ \frac {x^{2}+y^{2}}{2 x^{8}} = 2 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \sqrt {4 x^{6}-1}\, x \] |
Problem 1005
| ODE |
\[ \boxed {y^{\prime }-\frac {x +y}{x -y}=0} \] |
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program solution |
\[ \frac {\ln \left (x^{2}+y^{2}\right )}{2}+\arctan \left (\frac {x}{y}\right ) = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x \right )+2 c_{1} \right )\right ) x \] |
Problem 1006
| ODE |
\[ \boxed {\left (-y+y^{\prime } x \right ) \left (\ln \left (y\right )-\ln \left (x \right )\right )=x} \] |
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program solution |
\[ y = {\mathrm e}^{\operatorname {LambertW}\left (\left (\ln \left (x \right )+c_{1} \right ) {\mathrm e}^{-1}\right )+1} x \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x \ln \left (\frac {x}{c_{1}}\right )}{\operatorname {LambertW}\left (\ln \left (\frac {x}{c_{1}}\right ) {\mathrm e}^{-1}\right )} \] |
Problem 1007
| ODE |
\[ \boxed {y^{\prime }-\frac {y^{3}+2 x y^{2}+x^{2} y+x^{3}}{x \left (x +y\right )^{2}}=0} \] |
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program solution |
\[ \frac {y^{3}}{3 x^{3}}+\frac {y^{2}}{x^{2}}+\frac {y}{x}-\ln \left (x \right ) = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= x \left (-1+\left (3 \ln \left (x \right )+3 c_{1} \right )^{\frac {1}{3}}\right ) \\ y \left (x \right ) &= -\frac {x \left (i \sqrt {3}\, \left (3 \ln \left (x \right )+3 c_{1} \right )^{\frac {1}{3}}+\left (3 \ln \left (x \right )+3 c_{1} \right )^{\frac {1}{3}}+2\right )}{2} \\ y \left (x \right ) &= \frac {x \left (i \sqrt {3}-1\right ) \left (3 \ln \left (x \right )+3 c_{1} \right )^{\frac {1}{3}}}{2}-x \\ \end{align*} |
Problem 1008
| ODE |
\[ \boxed {y^{\prime }-\frac {x +2 y}{2 x +y}=0} \] |
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program solution |
\[ -\frac {3 \ln \left (x -y\right )}{2}+\frac {\ln \left (x +y\right )}{2} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {x \left (\frac {c_{1} \left (\left (27 c_{1} x +3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {2}{3}}+3\right )}{3 x \left (27 c_{1} x +3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}}}+c_{1}^{2}\right )}{c_{1}^{2}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (27 c_{1} x +3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {2}{3}}-6 x c_{1} \left (27 c_{1} x +3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}}-3 i \sqrt {3}+3}{6 \left (27 c_{1} x +3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (27 c_{1} x +3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {2}{3}}+6 x c_{1} \left (27 c_{1} x +3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}}-3 i \sqrt {3}-3}{6 \left (27 c_{1} x +3 \sqrt {3}\, \sqrt {27 c_{1}^{2} x^{2}-1}\right )^{\frac {1}{3}} c_{1}} \\ \end{align*} |
Problem 1009
| ODE |
\[ \boxed {y^{\prime }-\frac {y}{-2 x +y}=0} \] |
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program solution |
\[ -x y^{2}+\frac {y^{3}}{3} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {\left (-12 c_{1} +8 x^{3}+4 \sqrt {3}\, \sqrt {c_{1} \left (-4 x^{3}+3 c_{1} \right )}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{2}}{\left (-12 c_{1} +8 x^{3}+4 \sqrt {3}\, \sqrt {c_{1} \left (-4 x^{3}+3 c_{1} \right )}\right )^{\frac {1}{3}}}+x \\ y \left (x \right ) &= \frac {\left (-1-i \sqrt {3}\right ) \left (-12 c_{1} +8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}+\frac {x \left (i x \sqrt {3}-x +\left (-12 c_{1} +8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}\right )}{\left (-12 c_{1} +8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (-12 c_{1} +8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x \left (i x \sqrt {3}+x -\left (-12 c_{1} +8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}\right )}{\left (-12 c_{1} +8 x^{3}+4 \sqrt {3}\, \sqrt {-4 c_{1} x^{3}+3 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ \end{align*} |
Problem 1010
| ODE |
\[ \boxed {y^{\prime }-\frac {x y^{2}+2 y^{3}}{x^{3}+x^{2} y+x y^{2}}=0} \] |
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program solution |
\[ -\frac {3 \ln \left (-x +y\right )}{2}-\frac {\ln \left (x +y\right )}{2}+\ln \left (y\right ) = -2 \ln \left (x \right )+c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x \left (\operatorname {RootOf}\left (\textit {\_Z}^{8} c_{1} x^{2}+2 \textit {\_Z}^{6} c_{1} x^{2}+\textit {\_Z}^{4} c_{1} x^{2}-2 \textit {\_Z}^{2}-1\right )^{6} c_{1} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{8} c_{1} x^{2}+2 \textit {\_Z}^{6} c_{1} x^{2}+\textit {\_Z}^{4} c_{1} x^{2}-2 \textit {\_Z}^{2}-1\right )^{4} c_{1} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{8} c_{1} x^{2}+2 \textit {\_Z}^{6} c_{1} x^{2}+\textit {\_Z}^{4} c_{1} x^{2}-2 \textit {\_Z}^{2}-1\right )^{2} c_{1} x^{2}-1\right ) \] |
Problem 1011
| ODE |
\[ \boxed {y^{\prime }-\frac {x^{3}+x^{2} y+3 y^{3}}{x^{3}+3 x y^{2}}=0} \] |
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program solution |
\[ \frac {y^{3}}{x^{3}}+\frac {y}{x}-\ln \left (x \right ) = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {\left (\left (108 \ln \left (x \right )+108 c_{1} +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_{1} +81 c_{1}^{2}}\right )^{\frac {2}{3}}-12\right ) x}{6 \left (108 \ln \left (x \right )+108 c_{1} +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_{1} +81 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (i \sqrt {3}\, \left (108 \ln \left (x \right )+108 c_{1} +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_{1} +81 c_{1}^{2}}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (108 \ln \left (x \right )+108 c_{1} +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_{1} +81 c_{1}^{2}}\right )^{\frac {2}{3}}-12\right ) x}{12 \left (108 \ln \left (x \right )+108 c_{1} +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_{1} +81 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {x \left (\left (i \sqrt {3}-1\right ) \left (108 \ln \left (x \right )+108 c_{1} +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_{1} +81 c_{1}^{2}}\right )^{\frac {2}{3}}+12 i \sqrt {3}+12\right )}{12 \left (108 \ln \left (x \right )+108 c_{1} +12 \sqrt {12+81 \ln \left (x \right )^{2}+162 \ln \left (x \right ) c_{1} +81 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ \end{align*} |
Problem 1012
| ODE |
\[ \boxed {y^{\prime } x^{2}-y^{2}-y x=-4 x^{2}} \] With initial conditions \begin {align*} [y \left (-1\right ) = 0] \end {align*}
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program solution |
\[ y = -\frac {2 \left (x^{4}-1\right ) x}{x^{4}+1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-2 x^{5}+2 x}{x^{4}+1} \] |
Problem 1013
| ODE |
\[ \boxed {y^{\prime } x y-y^{2}+y x=x^{2}} \] |
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program solution |
\[ y = x \operatorname {LambertW}\left (\frac {{\mathrm e}^{c_{1} -1}}{x}\right )+x \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x \left (1+\operatorname {LambertW}\left (\frac {{\mathrm e}^{-1-c_{1}}}{x}\right )\right ) \] |
Problem 1014
| ODE |
\[ \boxed {y^{\prime }-\frac {2 y^{2}-y x +2 x^{2}}{y x +2 x^{2}}=0} \] |
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program solution |
\[ 3 \ln \left (-x +y\right )-4 \ln \left (-2 x +y\right ) = -2 \ln \left (x \right )+c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \operatorname {RootOf}\left (\textit {\_Z}^{4}+c_{1} x +16+\left (-3 c_{1} x -32\right ) \textit {\_Z} +\left (3 c_{1} x +24\right ) \textit {\_Z}^{2}+\left (-c_{1} x -8\right ) \textit {\_Z}^{3}\right ) x \] |
Problem 1015
| ODE |
\[ \boxed {y^{\prime }-\frac {x^{2}+y x +y^{2}}{y x}=0} \] |
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program solution |
\[ y = -x \operatorname {LambertW}\left (-\frac {{\mathrm e}^{c_{1} -1}}{x}\right )-x \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x \left (-\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-1-c_{1}}}{x}\right )-1\right ) \] |
Problem 1016
| ODE |
\[ \boxed {y^{\prime }-\frac {-6 x +y-3}{2 x -y-1}=0} \] |
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program solution |
\[ y = \left (x +1\right ) \operatorname {RootOf}\left (\left (x^{5}+5 x^{4}+10 x^{3}+10 x^{2}+5 x +1\right ) \textit {\_Z}^{5}+\left (5 x^{5}+25 x^{4}+50 x^{3}+50 x^{2}+25 x +5\right ) \textit {\_Z}^{4}+\left (-40 x^{5}-200 x^{4}-400 x^{3}-400 x^{2}-200 x -40\right ) \textit {\_Z}^{2}+\left (-80 x^{5}-400 x^{4}-800 x^{3}-800 x^{2}-400 x -80\right ) \textit {\_Z} -48 x^{5}-240 x^{4}-480 x^{3}-c_{4} {\mathrm e}^{5 c_{3}}-480 x^{2}-240 x -48\right )-3 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-\operatorname {RootOf}\left (\textit {\_Z}^{25}+\left (-5 c_{1} x^{5}-25 c_{1} x^{4}-50 c_{1} x^{3}-50 c_{1} x^{2}-25 c_{1} x -5 c_{1} \right ) \textit {\_Z}^{5}-c_{1} x^{5}-5 c_{1} x^{4}-10 c_{1} x^{3}-10 c_{1} x^{2}-5 c_{1} x -c_{1} \right )^{20}+3 c_{1} x \left (x +1\right )^{4}}{c_{1} \left (x +1\right )^{4}} \] |
Problem 1017
| ODE |
\[ \boxed {y^{\prime }-\frac {2 x +y+1}{x +2 y-4}=0} \] |
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program solution |
\[ y = \left (2+x \right ) \operatorname {RootOf}\left (\left (x^{4}+8 x^{3}+24 x^{2}+32 x +16\right ) \textit {\_Z}^{4}+\left (-2 x^{4}-16 x^{3}-48 x^{2}-64 x -32\right ) \textit {\_Z}^{3}+\left (2 x^{4}+16 x^{3}+48 x^{2}+64 x +32\right ) \textit {\_Z} -x^{4}-8 x^{3}-c_{4} {\mathrm e}^{2 c_{3}}-24 x^{2}-32 x -16\right )+3 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (x +5\right ) \operatorname {RootOf}\left (\textit {\_Z}^{16}+\left (2 c_{1} x^{4}+16 c_{1} x^{3}+48 c_{1} x^{2}+64 c_{1} x +32 c_{1} \right ) \textit {\_Z}^{4}-c_{1} x^{4}-8 c_{1} x^{3}-24 c_{1} x^{2}-32 c_{1} x -16 c_{1} \right )^{4}-2-x}{\operatorname {RootOf}\left (\textit {\_Z}^{16}+\left (2 c_{1} x^{4}+16 c_{1} x^{3}+48 c_{1} x^{2}+64 c_{1} x +32 c_{1} \right ) \textit {\_Z}^{4}-c_{1} x^{4}-8 c_{1} x^{3}-24 c_{1} x^{2}-32 c_{1} x -16 c_{1} \right )^{4}} \] |
Problem 1018
| ODE |
\[ \boxed {y^{\prime }-\frac {-x +3 y-14}{x +y-2}=0} \] |
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program solution |
\[ \ln \left (\frac {y-6-x}{2+x}\right )+\frac {4+2 x}{-y+6+x}+\ln \left (2+x \right )-c_{3} = 0 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (x +6\right ) \operatorname {LambertW}\left (-2 c_{1} \left (2+x \right )\right )+2 x +4}{\operatorname {LambertW}\left (-2 c_{1} \left (2+x \right )\right )} \] |
Problem 1019
| ODE |
\[ \boxed {3 y^{\prime } y^{2} x -y^{3}=x} \] |
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program solution |
\[ \frac {y^{3}}{x}-\ln \left (x \right ) = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \left (\left (\ln \left (x \right )+c_{1} \right ) x \right )^{\frac {1}{3}} \\ y \left (x \right ) &= -\frac {\left (\left (\ln \left (x \right )+c_{1} \right ) x \right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2} \\ y \left (x \right ) &= \frac {\left (\left (\ln \left (x \right )+c_{1} \right ) x \right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{2} \\ \end{align*} |
Problem 1020
| ODE |
\[ \boxed {y^{\prime } x y-6 y^{2}=3 x^{6}} \] |
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program solution |
\[ \frac {x^{6}+y^{2}}{2 x^{12}} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \sqrt {c_{1} x^{6}-1}\, x^{3} \\ y \left (x \right ) &= -\sqrt {c_{1} x^{6}-1}\, x^{3} \\ \end{align*} |
Problem 1021
| ODE |
\[ \boxed {y^{\prime } x^{3}-2 y^{2}-2 x^{2} y=-2 x^{4}} \] |
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program solution |
\[ y = -\frac {\left (c_{3} x^{4}-1\right ) x^{2}}{c_{3} x^{4}+1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \tanh \left (-2 \ln \left (x \right )+2 c_{1} \right ) x^{2} \] |
Problem 1022
| ODE |
\[ \boxed {y^{\prime }-y^{2} {\mathrm e}^{-x}-4 y=2 \,{\mathrm e}^{x}} \] |
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program solution |
\[ y = \frac {-2 c_{3} {\mathrm e}^{2 x}-{\mathrm e}^{x}}{{\mathrm e}^{x} c_{3} +1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {2 \,{\mathrm e}^{x} \left ({\mathrm e}^{x} c_{1} -1\right )}{-2+{\mathrm e}^{x} c_{1}} \] |
Problem 1023
| ODE |
\[ \boxed {y^{\prime }-\frac {y^{2}+\tan \left (x \right ) y+\tan \left (x \right )^{2}}{\sin \left (x \right )^{2}}=0} \] |
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program solution |
\[ y = \frac {\left (-\sin \left (\frac {\ln \left (\sin \left (x \right )+1\right )}{2}+\frac {\ln \left (\sin \left (x \right )-1\right )}{2}-\ln \left (\sin \left (x \right )\right )\right )+c_{3} \cos \left (\frac {\ln \left (\sin \left (x \right )+1\right )}{2}+\frac {\ln \left (\sin \left (x \right )-1\right )}{2}-\ln \left (\sin \left (x \right )\right )\right )\right ) \tan \left (x \right )}{c_{3} \sin \left (\frac {\ln \left (\sin \left (x \right )+1\right )}{2}+\frac {\ln \left (\sin \left (x \right )-1\right )}{2}-\ln \left (\sin \left (x \right )\right )\right )+\cos \left (\frac {\ln \left (\sin \left (x \right )+1\right )}{2}+\frac {\ln \left (\sin \left (x \right )-1\right )}{2}-\ln \left (\sin \left (x \right )\right )\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {\tan \left (x \right ) \left (c_{1} \sin \left (\frac {\ln \left (\sin \left (x \right )-1\right )}{2}+\frac {\ln \left (\sin \left (x \right )+1\right )}{2}-\ln \left (\sin \left (x \right )\right )\right )-\cos \left (\frac {\ln \left (\sin \left (x \right )-1\right )}{2}+\frac {\ln \left (\sin \left (x \right )+1\right )}{2}-\ln \left (\sin \left (x \right )\right )\right )\right )}{c_{1} \cos \left (\frac {\ln \left (\sin \left (x \right )-1\right )}{2}+\frac {\ln \left (\sin \left (x \right )+1\right )}{2}-\ln \left (\sin \left (x \right )\right )\right )+\sin \left (\frac {\ln \left (\sin \left (x \right )-1\right )}{2}+\frac {\ln \left (\sin \left (x \right )+1\right )}{2}-\ln \left (\sin \left (x \right )\right )\right )} \] |
Problem 1024
| ODE |
\[ \boxed {x \ln \left (x \right )^{2} y^{\prime }-\ln \left (x \right ) y-y^{2}=-4 \ln \left (x \right )^{2}} \] |
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program solution |
\[ y = -\frac {2 \left (\ln \left (x \right )^{4}-c_{3} \right ) \ln \left (x \right )}{\ln \left (x \right )^{4}+c_{3}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 2 i \tan \left (2 i \ln \left (\ln \left (x \right )\right )+c_{1} \right ) \ln \left (x \right ) \] |
Problem 1025
| ODE |
\[ \boxed {2 x \left (y+2 \sqrt {x}\right ) y^{\prime }-\left (y+\sqrt {x}\right )^{2}=0} \] |
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program solution |
\[ -\ln \left (x \right )+\frac {4 y}{\sqrt {x}}+\frac {y^{2}}{x} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {-2 x +\sqrt {x^{2} \left (\ln \left (x \right )-c_{1} +4\right )}}{\sqrt {x}} \\ y \left (x \right ) &= -\frac {2 x +\sqrt {x^{2} \left (\ln \left (x \right )-c_{1} +4\right )}}{\sqrt {x}} \\ \end{align*} |
Problem 1026
| ODE |
\[ \boxed {\left (y+{\mathrm e}^{x^{2}}\right ) y^{\prime }-2 x \left (y^{2}+{\mathrm e}^{x^{2}} y+{\mathrm e}^{2 x^{2}}\right )=0} \] |
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program solution |
\[ -x^{2}+\frac {{\mathrm e}^{-2 x^{2}} y^{2}}{2}+{\mathrm e}^{-x^{2}} y = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \left (-1-\sqrt {2 x^{2}-2 c_{1} +1}\right ) {\mathrm e}^{x^{2}} \\ y \left (x \right ) &= \left (-1+\sqrt {2 x^{2}-2 c_{1} +1}\right ) {\mathrm e}^{x^{2}} \\ \end{align*} |
Problem 1027
| ODE |
\[ \boxed {y^{\prime }+\frac {2 y}{x}-\frac {3 x^{2} y^{2}+6 y x +2}{x^{2} \left (2 y x +3\right )}=0} \] With initial conditions \begin {align*} [y \left (2\right ) = 2] \end {align*}
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program solution |
\[ \frac {x y^{2}}{2}+\frac {1}{x}+\frac {3 y}{2} = {\frac {15}{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-3+\sqrt {60 x +1}}{2 x} \] |
Problem 1028
| ODE |
\[ \boxed {y^{\prime }+\frac {3 y}{x}-\frac {3 x^{4} y^{2}+10 x^{2} y+6}{x^{3} \left (2 x^{2} y+5\right )}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
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program solution |
\[ y^{2} x^{3}+5 y x +\frac {6}{x} = 12 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-5+\sqrt {48 x +1}}{2 x^{2}} \] |
Problem 1029
| ODE |
\[ \boxed {y^{\prime }+\left (1+2 x \right ) y-x y^{2}=x +1} \] |
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program solution |
\[ y = \frac {c_{3} {\mathrm e}^{-\frac {x^{2}}{2}}+{\mathrm e}^{-\frac {x \left (2+x \right )}{2}} x +2 \,{\mathrm e}^{-\frac {x \left (2+x \right )}{2}}}{c_{3} {\mathrm e}^{-\frac {x^{2}}{2}}+{\mathrm e}^{-\frac {x \left (2+x \right )}{2}} x +{\mathrm e}^{-\frac {x \left (2+x \right )}{2}}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (2 x +4\right ) {\mathrm e}^{-x}-c_{1}}{\left (2+2 x \right ) {\mathrm e}^{-x}-c_{1}} \] |
Problem 1030
| ODE |
\[ \boxed {6 x^{2} y^{2}+4 y^{\prime } y x^{3}=0} \] |
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program solution |
\[ y = {\mathrm e}^{-\frac {3 \ln \left (x \right )}{2}-\frac {3 c_{1}}{2}} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {-c_{1} x}}{2 x^{2}} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {-c_{1} x}}{2 x^{2}} \\ \end{align*} |
Problem 1031
| ODE |
\[ \boxed {3 \cos \left (x \right ) y+2 y x^{3}+\left (3 \sin \left (x \right )+3\right ) y^{\prime }=-4 x \,{\mathrm e}^{x}} \] |
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program solution |
\[ \int _{}^{x}\left (2 y \textit {\_a}^{3}+4 \textit {\_a} \,{\mathrm e}^{\textit {\_a}}+3 \cos \left (\textit {\_a} \right ) y\right ) {\mathrm e}^{\frac {12 \textit {\_a}^{2} \left ({\mathrm e}^{i \textit {\_a}}+i\right ) \ln \left (1-i {\mathrm e}^{i \textit {\_a}}\right )+\left (-24 i {\mathrm e}^{i \textit {\_a}} \textit {\_a} +24 \textit {\_a} \right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i \textit {\_a}}\right )+\left (24 i+24 \,{\mathrm e}^{i \textit {\_a}}\right ) \operatorname {polylog}\left (3, i {\mathrm e}^{i \textit {\_a}}\right )-4 i {\mathrm e}^{i \textit {\_a}} \textit {\_a}^{3}}{3 \,{\mathrm e}^{i \textit {\_a}}+3 i}}d \textit {\_a} +\left (-\left (\int _{}^{x}\left (2 \textit {\_a}^{3}+3 \cos \left (\textit {\_a} \right )\right ) {\mathrm e}^{\frac {12 \textit {\_a}^{2} \left ({\mathrm e}^{i \textit {\_a}}+i\right ) \ln \left (1-i {\mathrm e}^{i \textit {\_a}}\right )+\left (-24 i {\mathrm e}^{i \textit {\_a}} \textit {\_a} +24 \textit {\_a} \right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i \textit {\_a}}\right )+\left (24 i+24 \,{\mathrm e}^{i \textit {\_a}}\right ) \operatorname {polylog}\left (3, i {\mathrm e}^{i \textit {\_a}}\right )-4 i {\mathrm e}^{i \textit {\_a}} \textit {\_a}^{3}}{3 \,{\mathrm e}^{i \textit {\_a}}+3 i}}d \textit {\_a} \right )+\left (3 \sin \left (x \right )+3\right ) {\mathrm e}^{\frac {12 x^{2} \left ({\mathrm e}^{i x}+i\right ) \ln \left (1-i {\mathrm e}^{i x}\right )+\left (-24 i {\mathrm e}^{i x} x +24 x \right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )+\left (24 i+24 \,{\mathrm e}^{i x}\right ) \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )-4 i {\mathrm e}^{i x} x^{3}}{3 \,{\mathrm e}^{i x}+3 i}}\right ) y = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {{\mathrm e}^{\frac {\left (-24 x +24 i x \,{\mathrm e}^{i x}\right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )+\left (-24 i-24 \,{\mathrm e}^{i x}\right ) \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )+4 i x \left (x^{2}+\frac {3}{4}\right ) {\mathrm e}^{i x}-3 x}{3 \,{\mathrm e}^{i x}+3 i}} \left (4 \left (\int \frac {\left (-2 i {\mathrm e}^{\frac {24 x \left ({\mathrm e}^{i x}+i\right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )+4 \,{\mathrm e}^{i x} x^{3}+3 i x \,{\mathrm e}^{i x}+24 i \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right ) {\mathrm e}^{i x}-3 x -24 \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )}{3 i {\mathrm e}^{i x}-3}}+{\mathrm e}^{\frac {\left (24 i+24 \,{\mathrm e}^{i x}\right ) \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )-4 x \left (\left (6 i {\mathrm e}^{i x}-6\right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )+\left (i x^{2}-\frac {3}{4}+\frac {3}{4} i\right ) {\mathrm e}^{i x}-\frac {3}{4}-\frac {3 i}{4}\right )}{3 \,{\mathrm e}^{i x}+3 i}}-{\mathrm e}^{\frac {\left (24 i+24 \,{\mathrm e}^{i x}\right ) \operatorname {polylog}\left (3, i {\mathrm e}^{i x}\right )-4 x \left (\left (6 i {\mathrm e}^{i x}-6\right ) \operatorname {polylog}\left (2, i {\mathrm e}^{i x}\right )+\left (i x^{2}-\frac {3}{4}-\frac {3}{4} i\right ) {\mathrm e}^{i x}+\frac {3}{4}-\frac {3 i}{4}\right )}{3 \,{\mathrm e}^{i x}+3 i}}\right ) \left (1-i {\mathrm e}^{i x}\right )^{4 x^{2}} x}{\sin \left (x \right )+1}d x \right )+3 c_{1} \right ) \left (1-i {\mathrm e}^{i x}\right )^{-4 x^{2}}}{3 \left ({\mathrm e}^{i x}+i\right )^{2}} \] |
Problem 1032
| ODE |
\[ \boxed {14 y^{3} x^{2}+21 x^{2} y^{2} y^{\prime }=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-\frac {2 x}{3}} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= c_{1} {\mathrm e}^{-\frac {2 x}{3}} \\ \end{align*} |
Problem 1033
| ODE |
\[ \boxed {-2 y^{2}+\left (12 y^{2}-4 y x \right ) y^{\prime }=-2 x} \] |
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program solution |
\[ x \left (-2 y^{2}+x \right )+4 y^{3} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {\left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-6 x^{5}-6 c_{1} x^{3}+81 x^{4}+162 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{6}+\frac {x^{2}}{6 \left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-6 x^{5}-6 c_{1} x^{3}+81 x^{4}+162 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}+\frac {x}{6} \\ y \left (x \right ) &= \frac {\left (-\left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-3 \left (x^{2}+c_{1} \right ) \left (2 x^{3}-27 x^{2}-27 c_{1} \right )}\right )^{\frac {1}{3}}+x \right ) \left (i \left (\left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-3 \left (x^{2}+c_{1} \right ) \left (2 x^{3}-27 x^{2}-27 c_{1} \right )}\right )^{\frac {1}{3}}+x \right ) \sqrt {3}+\left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-3 \left (x^{2}+c_{1} \right ) \left (2 x^{3}-27 x^{2}-27 c_{1} \right )}\right )^{\frac {1}{3}}-x \right )}{12 \left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-3 \left (x^{2}+c_{1} \right ) \left (2 x^{3}-27 x^{2}-27 c_{1} \right )}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-3 \left (x^{2}+c_{1} \right ) \left (2 x^{3}-27 x^{2}-27 c_{1} \right )}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{12}-\frac {x \left (i x \sqrt {3}+x -2 \left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-3 \left (x^{2}+c_{1} \right ) \left (2 x^{3}-27 x^{2}-27 c_{1} \right )}\right )^{\frac {1}{3}}\right )}{12 \left (-27 x^{2}-27 c_{1} +x^{3}+3 \sqrt {-3 \left (x^{2}+c_{1} \right ) \left (2 x^{3}-27 x^{2}-27 c_{1} \right )}\right )^{\frac {1}{3}}} \\ \end{align*} |
Problem 1034
| ODE |
\[ \boxed {\left (x +y\right )^{2}+\left (x +y\right )^{2} y^{\prime }=0} \] |
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program solution |
\[ y = -x +c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -x \\ y \left (x \right ) &= c_{1} -x \\ y \left (x \right ) &= -\frac {c_{1}}{2}-\frac {i \sqrt {3}\, c_{1}}{2}-x \\ y \left (x \right ) &= -\frac {c_{1}}{2}+\frac {i \sqrt {3}\, c_{1}}{2}-x \\ \end{align*} |
Problem 1035
| ODE |
\[ \boxed {7 y+\left (3 x +4 y\right ) y^{\prime }=-4 x} \] |
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program solution |
\[ \frac {5 \ln \left (2 x +y\right )}{6}+\frac {\ln \left (x +2 y\right )}{6} = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{36}+3 \textit {\_Z}^{6} c_{1} x^{6}-2 c_{1} x^{6}\right )^{6}+1\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{36}+3 \textit {\_Z}^{6} c_{1} x^{6}-2 c_{1} x^{6}\right )^{6}} \] |
Problem 1036
| ODE |
\[ \boxed {-2 \sin \left (x \right ) y^{2}+3 y^{3}+\left (4 \cos \left (x \right ) y+9 x y^{2}\right ) y^{\prime }=2 x} \] |
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program solution |
\[ 3 x y^{3}-x^{2}+2 \cos \left (x \right ) y^{2} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {\frac {\left (972 x^{4}+36 \sqrt {3}\, \sqrt {\left (-x^{2}+c_{1} \right ) \left (-243 x^{4}+32 \cos \left (x \right )^{3}+243 c_{1} x^{2}\right )}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}}{2}+\frac {8 \cos \left (x \right )^{2}}{\left (972 x^{4}+36 \sqrt {3}\, \sqrt {\left (-x^{2}+c_{1} \right ) \left (-243 x^{4}+32 \cos \left (x \right )^{3}+243 c_{1} x^{2}\right )}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}}-2 \cos \left (x \right )}{9 x} \\ y \left (x \right ) &= -\frac {\left (\left (972 x^{4}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_{1} \right )+243 \left (x^{2}-c_{1} \right )^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}+4 \cos \left (x \right )\right ) \left (i \left (\left (972 x^{4}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_{1} \right )+243 \left (x^{2}-c_{1} \right )^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}-4 \cos \left (x \right )\right ) \sqrt {3}+\left (972 x^{4}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_{1} \right )+243 \left (x^{2}-c_{1} \right )^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}+4 \cos \left (x \right )\right )}{36 x \left (972 x^{4}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_{1} \right )+243 \left (x^{2}-c_{1} \right )^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \left (\left (972 x^{4}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_{1} \right )+243 \left (x^{2}-c_{1} \right )^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}-4 \cos \left (x \right )\right ) \sqrt {3}-\left (972 x^{4}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_{1} \right )+243 \left (x^{2}-c_{1} \right )^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}-4 \cos \left (x \right )\right ) \left (\left (972 x^{4}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_{1} \right )+243 \left (x^{2}-c_{1} \right )^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}+4 \cos \left (x \right )\right )}{36 x \left (972 x^{4}+36 \sqrt {3}\, \sqrt {32 \cos \left (x \right )^{3} \left (-x^{2}+c_{1} \right )+243 \left (x^{2}-c_{1} \right )^{2} x^{2}}\, x -64 \cos \left (x \right )^{3}-972 c_{1} x^{2}\right )^{\frac {1}{3}}} \\ \end{align*} |
Problem 1037
| ODE |
\[ \boxed {y+\left (2 y+2 x \right ) y^{\prime }=-2 x} \] |
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program solution |
\[ \frac {\ln \left (2 y^{2}+3 y x +2 x^{2}\right )}{2}+\frac {\sqrt {7}\, \arctan \left (\frac {\left (3 x +4 y\right ) \sqrt {7}}{7 x}\right )}{7} = c_{1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x \left (\sqrt {7}\, \tan \left (\operatorname {RootOf}\left (\sqrt {7}\, \ln \left (\sec \left (\textit {\_Z} \right )^{2} x^{2}\right )+\sqrt {7}\, \ln \left (7\right )-3 \sqrt {7}\, \ln \left (2\right )+2 \sqrt {7}\, c_{1} +2 \textit {\_Z} \right )\right )-3\right )}{4} \] |
Problem 1038
| ODE |
\[ \boxed {2 y x +4 y^{2}+\left (x^{2}+8 y x +18 y\right ) y^{\prime }=-3 x^{2}} \] |
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program solution |
\[ x^{3}+x^{2} y+4 x y^{2}+9 y^{2} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {-x^{2}+\sqrt {-15 x^{4}-36 x^{3}-16 c_{1} x -36 c_{1}}}{8 x +18} \\ y \left (x \right ) &= \frac {-x^{2}-\sqrt {-15 x^{4}-36 x^{3}-16 c_{1} x -36 c_{1}}}{8 x +18} \\ \end{align*} |
Problem 1039
| ODE |
\[ \boxed {8 y x +y^{2}+\left (2 x^{2}+\frac {x y^{3}}{3}\right ) y^{\prime }=-2 x^{2}} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 1040
| ODE |
\[ \boxed {\left (\frac {1}{y}+2 y\right ) y^{\prime }=-\frac {1}{x}-2 x} \] |
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program solution |
\[ y = \frac {{\mathrm e}^{-\frac {\operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{-2 x^{2}-2 c_{1}}}{x^{2}}\right )}{2}-x^{2}-c_{1}}}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {{\mathrm e}^{-x^{2}-c_{1}} \sqrt {2}}{2 \sqrt {\frac {{\mathrm e}^{-2 x^{2}-2 c_{1}}}{x^{2} \operatorname {LambertW}\left (\frac {2 \,{\mathrm e}^{-2 x^{2}-2 c_{1}}}{x^{2}}\right )}}\, x} \] |
Problem 1041
| ODE |
\[ \boxed {y \sin \left (x y\right )+x y^{2} \cos \left (x y\right )+\left (x \sin \left (x y\right )+x y^{2} \cos \left (x y\right )\right ) y^{\prime }=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 1042
| ODE |
\[ \boxed {\frac {x}{\left (x^{2}+y^{2}\right )^{\frac {3}{2}}}+\frac {y y^{\prime }}{\left (x^{2}+y^{2}\right )^{\frac {3}{2}}}=0} \] |
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program solution |
\[ -\frac {x^{2}}{2}-\frac {y^{2}}{2} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \sqrt {-x^{2}+c_{1}} \\ y \left (x \right ) &= -\sqrt {-x^{2}+c_{1}} \\ \end{align*} |
Problem 1043
| ODE |
\[ \boxed {{\mathrm e}^{x} \left (x^{2} y^{2}+2 x y^{2}\right )+\left (2 x^{2} y \,{\mathrm e}^{x}+2\right ) y^{\prime }=-6 x} \] |
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program solution |
\[ x^{2} \left (3+y^{2} {\mathrm e}^{x}\right )+2 y = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {\left (-1+\sqrt {-3 \,{\mathrm e}^{x} x^{4}-{\mathrm e}^{x} c_{1} x^{2}+1}\right ) {\mathrm e}^{-x}}{x^{2}} \\ y \left (x \right ) &= \frac {\left (-1-\sqrt {-3 \,{\mathrm e}^{x} x^{4}-{\mathrm e}^{x} c_{1} x^{2}+1}\right ) {\mathrm e}^{-x}}{x^{2}} \\ \end{align*} |
Problem 1044
| ODE |
\[ \boxed {x^{2} {\mathrm e}^{x^{2}+y} \left (2 x^{2}+3\right )+\left (x^{3} {\mathrm e}^{x^{2}+y}-12 y^{2}\right ) y^{\prime }=-4 x} \] |
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program solution |
\[ x^{2} \left ({\mathrm e}^{x^{2}+y} x +2\right )-4 y^{3} = c_{1} \] Verified OK.
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Maple solution | \[ x^{3} {\mathrm e}^{x^{2}+y \left (x \right )}-4 y \left (x \right )^{3}+2 x^{2}+c_{1} = 0 \] |
Problem 1045
| ODE |
\[ \boxed {{\mathrm e}^{y x} \left (y x^{4}+4 x^{3}\right )+3 y+\left (x^{5} {\mathrm e}^{y x}+3 x \right ) y^{\prime }=0} \] |
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program solution |
\[ y = -\frac {3 \operatorname {LambertW}\left (\frac {x^{4} {\mathrm e}^{\frac {c_{1}}{3}}}{3}\right )-c_{1}}{3 x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-3 \operatorname {LambertW}\left (\frac {x^{4} {\mathrm e}^{-\frac {c_{1}}{3}}}{3}\right )-c_{1}}{3 x} \] |
Problem 1046
| ODE |
\[ \boxed {3 y \cos \left (x \right ) x^{2}-x^{3} y^{2} \sin \left (x \right )+\left (8 y-\sin \left (x \right ) y x^{4}\right ) y^{\prime }=-4 x} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 1047
| ODE |
\[ \boxed {4 y^{2} x^{3}-6 x^{2} y+\left (2 y x^{4}-2 x^{3}\right ) y^{\prime }=2 x +3} \] With initial conditions \begin {align*} [y \left (1\right ) = 3] \end {align*}
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program solution |
\[ x^{4} y^{2}-2 y x^{3}-x^{2}-3 x = -1 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x +\sqrt {2 x^{2}+3 x -1}}{x^{2}} \] |
Problem 1048
| ODE |
\[ \boxed {-4 \cos \left (x \right ) y+\left (4 y-4 \sin \left (x \right )\right ) y^{\prime }=-4 \sin \left (x \right ) \cos \left (x \right )-\sec \left (x \right )^{2}} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{4}\right ) = 0\right ] \end {align*}
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program solution |
\[ \tan \left (x \right )-4 \sin \left (x \right ) y+2 \sin \left (x \right )^{2}+2 y^{2} = 2 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \sin \left (x \right )-\frac {\sec \left (x \right )^{2} \sqrt {2}\, \sqrt {\cos \left (x \right )^{3} \left (2 \cos \left (x \right )-\sin \left (x \right )\right )}}{2} \] |
Problem 1049
| ODE |
\[ \boxed {\left (y^{3}-1\right ) {\mathrm e}^{x}+3 y^{2} \left (1+{\mathrm e}^{x}\right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
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program solution |
\[ -\ln \left (1+{\mathrm e}^{x}\right )-\ln \left (y^{3}-1\right ) = -\ln \left (2\right )-i \pi \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {{\left (\left ({\mathrm e}^{x}-1\right ) \left (1+{\mathrm e}^{x}\right )^{2}\right )}^{\frac {1}{3}}}{1+{\mathrm e}^{x}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) {\left (\left ({\mathrm e}^{x}-1\right ) \left (1+{\mathrm e}^{x}\right )^{2}\right )}^{\frac {1}{3}}}{2+2 \,{\mathrm e}^{x}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) {\left (\left ({\mathrm e}^{x}-1\right ) \left (1+{\mathrm e}^{x}\right )^{2}\right )}^{\frac {1}{3}}}{2+2 \,{\mathrm e}^{x}} \\ \end{align*} |
Problem 1050
| ODE |
\[ \boxed {-\sin \left (x \right ) y+y^{\prime } \cos \left (x \right )=-\sin \left (x \right )+2 \cos \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
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program solution |
\[ y = 1+2 \tan \left (x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = 2 \tan \left (x \right )+1 \] |
Problem 1051
| ODE |
\[ \boxed {\left (2 x -1\right ) \left (y-1\right )+\left (2+x \right ) \left (x -3\right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -1] \end {align*}
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program solution |
\[ y = \frac {x^{2}-x +6}{x^{2}-x -6} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x^{2}-x +6}{\left (2+x \right ) \left (x -3\right )} \] |
Problem 1052
| ODE |
\[ \boxed {4 y+\left (4 x +3 y\right ) y^{\prime }=-7 x} \] |
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program solution |
\[ \frac {x \left (7 x +8 y\right )}{2}+\frac {3 y^{2}}{2} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {-4 c_{1} x -\sqrt {-5 c_{1}^{2} x^{2}+3}}{3 c_{1}} \\ y \left (x \right ) &= \frac {-4 c_{1} x +\sqrt {-5 c_{1}^{2} x^{2}+3}}{3 c_{1}} \\ \end{align*} |
Problem 1053
| ODE |
\[ \boxed {{\mathrm e}^{x} \left (x^{4} y^{2}+4 y^{2} x^{3}+1\right )+\left (2 x^{4} y \,{\mathrm e}^{x}+2 y\right ) y^{\prime }=0} \] |
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program solution |
\[ \left (x^{4} y^{2}+1\right ) {\mathrm e}^{x}+y^{2} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {\sqrt {\left ({\mathrm e}^{x} x^{4}+1\right ) \left (-{\mathrm e}^{x}+c_{1} \right )}}{{\mathrm e}^{x} x^{4}+1} \\ y \left (x \right ) &= -\frac {\sqrt {\left ({\mathrm e}^{x} x^{4}+1\right ) \left (-{\mathrm e}^{x}+c_{1} \right )}}{{\mathrm e}^{x} x^{4}+1} \\ \end{align*} |
Problem 1054
| ODE |
\[ \boxed {x^{3} y^{4}+\left (x^{4} y^{3}+y\right ) y^{\prime }=-x} \] |
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program solution |
\[ \frac {\left (x^{2} y^{4}+1\right )^{2}}{4 y^{4}}+\frac {y^{2}}{2}-\frac {1}{4 y^{4}} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {\sqrt {-1-\sqrt {-2 x^{6}-4 c_{1} x^{4}+1}}}{x^{2}} \\ y \left (x \right ) &= \frac {\sqrt {-1+\sqrt {-2 x^{6}-4 c_{1} x^{4}+1}}}{x^{2}} \\ y \left (x \right ) &= -\frac {\sqrt {-1-\sqrt {-2 x^{6}-4 c_{1} x^{4}+1}}}{x^{2}} \\ y \left (x \right ) &= -\frac {\sqrt {-1+\sqrt {-2 x^{6}-4 c_{1} x^{4}+1}}}{x^{2}} \\ \end{align*} |
Problem 1055
| ODE |
\[ \boxed {2 y+\left (2 y+2 x \right ) y^{\prime }=-3 x^{2}} \] |
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program solution |
\[ x \left (x^{2}+2 y\right )+y^{2} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= -x -\sqrt {-x^{3}+x^{2}-c_{1}} \\ y \left (x \right ) &= -x +\sqrt {-x^{3}+x^{2}-c_{1}} \\ \end{align*} |
Problem 1056
| ODE |
\[ \boxed {x^{3} y^{4}+\left (x^{4} y^{3}+3 y\right ) y^{\prime }=-2 x} \] |
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program solution |
\[ \frac {\left (x^{2} y^{4}+2\right )^{2}}{4 y^{4}}+\frac {3 y^{2}}{2}-\frac {1}{y^{4}} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {\sqrt {-3-\sqrt {-4 x^{6}-4 c_{1} x^{4}+9}}}{x^{2}} \\ y \left (x \right ) &= \frac {\sqrt {-3+\sqrt {-4 x^{6}-4 c_{1} x^{4}+9}}}{x^{2}} \\ y \left (x \right ) &= -\frac {\sqrt {-3-\sqrt {-4 x^{6}-4 c_{1} x^{4}+9}}}{x^{2}} \\ y \left (x \right ) &= -\frac {\sqrt {-3+\sqrt {-4 x^{6}-4 c_{1} x^{4}+9}}}{x^{2}} \\ \end{align*} |
Problem 1057
| ODE |
\[ \boxed {y^{2}+2 y^{\prime } x y=-x^{2}} \] |
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program solution |
\[ \frac {x^{3}}{3}+x y^{2} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} 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 1058
| ODE |
\[ \boxed {y^{\prime }+\frac {2 y}{x}+\frac {2 x y}{x^{2}+2 x^{2} y+1}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -2] \end {align*}
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program solution |
\[ \frac {1}{4}+\left (y^{2}+y\right ) x^{4}+x^{2} y = {\frac {1}{4}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-x^{2}-1}{x^{2}} \] |
Problem 1059
| ODE |
\[ \boxed {y^{\prime }-\frac {3 y}{x}-\frac {2 x^{4} \left (4 x^{3}-3 y\right )}{3 x^{5}+3 x^{3}+2 y}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
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program solution |
\[ -4 x^{2}+\frac {3 y}{x^{3}}+\frac {y^{2}}{x^{6}}+\frac {3 y}{x} = 3 \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {\left (-3 x^{2}+\sqrt {9 x^{4}+34 x^{2}+21}-3\right ) x^{3}}{2} \] |
Problem 1060
| ODE |
\[ \boxed {2 y x +y^{\prime }+\frac {{\mathrm e}^{-x^{2}} \left (3 x +2 \,{\mathrm e}^{x^{2}} y\right )}{2 x +3 \,{\mathrm e}^{x^{2}} y}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1] \end {align*}
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program solution |
\[ \frac {3 x^{2}}{2}+2 y x \,{\mathrm e}^{x^{2}}+\frac {3 \,{\mathrm e}^{2 x^{2}} y^{2}}{2} = {\frac {3}{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -\frac {\left (2 x \,{\mathrm e}^{x^{2}}+\sqrt {{\mathrm e}^{2 x^{2}} \left (-5 x^{2}+9\right )}\right ) {\mathrm e}^{-2 x^{2}}}{3} \] |
Problem 1061
| ODE |
\[ \boxed {y+\left (2 x +\frac {1}{y}\right ) y^{\prime }=0} \] |
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program solution |
\[ y+x y^{2} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {-1+\sqrt {4 c_{1} x +1}}{2 x} \\ y \left (x \right ) &= \frac {-1-\sqrt {4 c_{1} x +1}}{2 x} \\ \end{align*} |
Problem 1062
| ODE |
\[ \boxed {-y^{2}+y^{\prime } x^{2}=0} \] |
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program solution |
\[ y = \frac {1}{c_{3} +\frac {1}{x}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {x}{c_{1} x +1} \] |
Problem 1063
| ODE |
\[ \boxed {-y^{\prime } x +y=0} \] |
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program solution |
\[ y = x \,{\mathrm e}^{c_{1}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} x \] |
Problem 1064
| ODE |
\[ \boxed {3 x^{2} y+2 y^{\prime } x^{3}=0} \] |
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program solution |
\[ y = {\mathrm e}^{-\frac {3 \ln \left (x \right )}{2}-\frac {3 c_{1}}{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1}}{x^{\frac {3}{2}}} \] |
Problem 1065
| ODE |
\[ \boxed {2 y^{3}+3 y^{2} y^{\prime }=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-\frac {2 x}{3}} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= c_{1} {\mathrm e}^{-\frac {2 x}{3}} \\ \end{align*} |
Problem 1066
| ODE |
\[ \boxed {5 y x +2 y+2 y^{\prime } x=-5} \] |
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program solution |
\[ y = -\frac {\left (2 \,{\mathrm e}^{\frac {5 x}{2}}-c_{1} \right ) {\mathrm e}^{-\frac {5 x}{2}}}{2 x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {{\mathrm e}^{-\frac {5 x}{2}} c_{1} -1}{x} \] |
Problem 1067
| ODE |
\[ \boxed {y x +2 y+\left (x +1\right ) y^{\prime }=-x -1} \] |
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program solution |
\[ y = -\frac {\left (x \,{\mathrm e}^{x}-c_{1} \right ) {\mathrm e}^{-x}}{x +1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {{\mathrm e}^{-x} c_{1} -x}{x +1} \] |
Problem 1068
| ODE |
\[ \boxed {27 x y^{2}+8 y^{3}+\left (18 x^{2} y+12 x y^{2}\right ) y^{\prime }=0} \] |
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program solution |
\[ y^{2} x^{2} \left (4 y+9 x \right ) = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {x \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}{3 c_{1} x^{5}+\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {x \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}{3 c_{1} x^{5}+\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {x \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}{3 c_{1} x^{5}+\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {x \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}{3 c_{1} x^{5}+\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= \frac {x \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}{3 c_{1} x^{5}+\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {2 x \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}{i \left (-3 c_{1} x^{5}+\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}\right ) \sqrt {3}+3 c_{1} x^{5}+\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {2 x \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}{\left (1-i \sqrt {3}\right ) \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}+3 c_{1} \left (1+i \sqrt {3}\right ) x^{5}} \\ y \left (x \right ) &= -\frac {2 x \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}{i \left (-3 c_{1} x^{5}+\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}\right ) \sqrt {3}+3 c_{1} x^{5}+\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {2 x \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}{\left (1-i \sqrt {3}\right ) \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}+3 c_{1} \left (1+i \sqrt {3}\right ) x^{5}} \\ y \left (x \right ) &= -\frac {2 x \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}{i \left (-3 c_{1} x^{5}+\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}\right ) \sqrt {3}+3 c_{1} x^{5}+\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {2 x \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}{\left (1-i \sqrt {3}\right ) \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}+3 c_{1} \left (1+i \sqrt {3}\right ) x^{5}} \\ y \left (x \right ) &= -\frac {2 x \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}{i \left (-3 c_{1} x^{5}+\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}\right ) \sqrt {3}+3 c_{1} x^{5}+\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {2 x \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}{\left (1-i \sqrt {3}\right ) \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}+3 c_{1} \left (1+i \sqrt {3}\right ) x^{5}} \\ y \left (x \right ) &= -\frac {2 x \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}{i \left (-3 c_{1} x^{5}+\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}\right ) \sqrt {3}+3 c_{1} x^{5}+\left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}} \\ y \left (x \right ) &= -\frac {2 x \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {1}{3}}}{\left (1-i \sqrt {3}\right ) \left (2 c_{1} x^{5}+\sqrt {-27 c_{1}^{3} x^{15}+4 c_{1}^{2} x^{10}}\right )^{\frac {2}{3}}+3 c_{1} \left (1+i \sqrt {3}\right ) x^{5}} \\ \end{align*} |
Problem 1069
| ODE |
\[ \boxed {6 x y^{2}+2 y+\left (12 x^{2} y+6 x +3\right ) y^{\prime }=0} \] |
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program solution |
\[ 3 x^{2} y^{4}+2 x y^{3}+y^{3} = c_{1} \] Verified OK.
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Maple solution | \[ 3 x^{2} y^{4}+\left (2 x +1\right ) y^{3}+c_{1} = 0 \] |
Problem 1070
| ODE |
\[ \boxed {y^{2}+\left (x y^{2}+6 y x +\frac {1}{y}\right ) y^{\prime }=0} \] |
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program solution |
\[ y^{6} {\mathrm e}^{y} x +\left (y^{3}-3 y^{2}+6 y-6\right ) {\mathrm e}^{y} = c_{1} \] Verified OK.
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Maple solution | \[ \frac {x y \left (x \right )^{6}+y \left (x \right )^{3}-3 y \left (x \right )^{2}-{\mathrm e}^{-y \left (x \right )} c_{1} +6 y \left (x \right )-6}{y \left (x \right )^{6}} = 0 \] |
Problem 1071
| ODE |
\[ \boxed {12 y x^{3}+24 x^{2} y^{2}+\left (9 x^{4}+32 y x^{3}+4 y\right ) y^{\prime }=0} \] |
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program solution |
\[ y^{3} x^{3} \left (3 x +8 y\right )+y^{4} = c_{1} \] Verified OK.
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Maple solution | \[ 3 x^{4} y \left (x \right )^{3}+8 x^{3} y \left (x \right )^{4}+y \left (x \right )^{4}+c_{1} = 0 \] |
Problem 1072
| ODE |
\[ \boxed {x^{2} y+4 y x +2 y+\left (x^{2}+x \right ) y^{\prime }=0} \] |
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program solution |
\[ y = \frac {{\mathrm e}^{-x -c_{1}}}{\left (x +1\right ) x^{2}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{-x}}{\left (x +1\right ) x^{2}} \] |
Problem 1073
| ODE |
\[ \boxed {-y+\left (x^{4}-x \right ) y^{\prime }=0} \] |
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program solution |
\[ y = \frac {{\mathrm e}^{\frac {\ln \left (x^{2}+x +1\right )}{3}+\frac {\ln \left (x -1\right )}{3}+c_{1}}}{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} \left (x -1\right )^{\frac {1}{3}} \left (x^{2}+x +1\right )^{\frac {1}{3}}}{x} \] |
Problem 1074
| ODE |
\[ \boxed {\cos \left (x \right ) \cos \left (y\right )+\left (\cos \left (y\right ) \sin \left (x \right )-\sin \left (y\right ) \sin \left (x \right )+y\right ) y^{\prime }=0} \] |
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program solution |
\[ \sin \left (x \right ) \cos \left (y\right ) {\mathrm e}^{y}+\left (y-1\right ) {\mathrm e}^{y} = c_{1} \] Verified OK.
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Maple solution | \[ \left (\sin \left (x \right ) \cos \left (y \left (x \right )\right )+y \left (x \right )-1\right ) {\mathrm e}^{y \left (x \right )}+c_{1} = 0 \] |
Problem 1075
| ODE |
\[ \boxed {2 y x +y^{2}+\left (2 y x +x^{2}-2 x y^{2}-2 x y^{3}\right ) y^{\prime }=0} \] |
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program solution |
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Maple solution | \[ \text {No solution found} \] |
Problem 1076
| ODE |
\[ \boxed {y \sin \left (y\right )+x \left (\sin \left (y\right )-y \cos \left (y\right )\right ) y^{\prime }=0} \] |
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program solution |
\[ -\ln \left (x \right )+\ln \left (\sin \left (y\right )\right )-\ln \left (y\right ) = c_{1} \] Verified OK.
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Maple solution | \[ \ln \left (x \right )+\ln \left (y \left (x \right )\right )-\ln \left (\sin \left (y \left (x \right )\right )\right )+c_{1} = 0 \] |
Problem 1077
| ODE |
\[ \boxed {a y+b x y+\left (c x +d x y\right ) y^{\prime }=0} \] |
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program solution |
\[ y = {\mathrm e}^{-\frac {a \ln \left (x \right )+b x +c \operatorname {LambertW}\left (\frac {d \,{\mathrm e}^{-\frac {a \ln \left (x \right )+b x +c_{1}}{c}}}{c}\right )+c_{1}}{c}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x^{-\frac {a}{c}} {\mathrm e}^{\frac {-b x -c \operatorname {LambertW}\left (\frac {d \,x^{-\frac {a}{c}} {\mathrm e}^{\frac {-b x -c_{1}}{c}}}{c}\right )-c_{1}}{c}} \] |
Problem 1078
| ODE |
\[ \boxed {3 y^{3} x^{2}-y^{2}+y+\left (-y x +2 x \right ) y^{\prime }=0} \] |
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program solution |
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {4}{\sqrt {x}\, \sqrt {\frac {c_{1} x +48 x^{2}+4}{x}}+2} \\ y \left (x \right ) &= -\frac {4}{\sqrt {x}\, \sqrt {\frac {c_{1} x +48 x^{2}+4}{x}}-2} \\ \end{align*} |
Problem 1079
| ODE |
\[ \boxed {2 y+3 \left (x^{2}+y^{3} x^{2}\right ) y^{\prime }=0} \] |
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program solution |
\[ y = {\mathrm e}^{-\frac {x \operatorname {LambertW}\left ({\mathrm e}^{-\frac {2 \left (c_{1} x -1\right )}{x}}\right )+2 c_{1} x -2}{3 x}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-\frac {x \operatorname {LambertW}\left ({\mathrm e}^{\frac {-2 c_{1} x +2}{x}}\right )+2 c_{1} x -2}{3 x}} \] |
Problem 1080
| ODE |
\[ \boxed {a \cos \left (x \right ) y-\sin \left (x \right ) y^{2}+\left (b \cos \left (x \right ) y-x \sin \left (x \right ) y\right ) y^{\prime }=0} \] |
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program solution |
\[ \int _{}^{x}\left (-\cos \left (\textit {\_a} \right ) a +\sin \left (\textit {\_a} \right ) y\right ) {\mathrm e}^{-i \textit {\_a} +2 \left (\int \frac {i \textit {\_a} -b}{\left (i b -\textit {\_a} \right ) {\mathrm e}^{2 i \textit {\_a}}+i b +\textit {\_a}}d \textit {\_a} \right )}d \textit {\_a} +\int _{0}^{y}\left (\sin \left (x \right ) {\mathrm e}^{-i x +2 \left (\int \frac {i x -b}{\left (i b -x \right ) {\mathrm e}^{2 i x}+i b +x}d x \right )} x -\cos \left (x \right ) {\mathrm e}^{-i x +2 \left (\int \frac {i x -b}{\left (i b -x \right ) {\mathrm e}^{2 i x}+i b +x}d x \right )} b -\left (\int _{}^{x}\sin \left (\textit {\_a} \right ) {\mathrm e}^{-i \textit {\_a} +2 \left (\int \frac {i b +\textit {\_a}}{{\mathrm e}^{2 i \textit {\_a}} \left (i \textit {\_a} +b \right )+b -i \textit {\_a}}d \textit {\_a} \right )}d \textit {\_a} \right )\right )d \textit {\_a} = c_{1} \] Warning, solution could not be verified |
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Maple solution | \begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \left (a \left (\int \frac {\cos \left (x \right ) {\mathrm e}^{\int \frac {\sin \left (x \right )}{x \sin \left (x \right )-\cos \left (x \right ) b}d x}}{x \sin \left (x \right )-\cos \left (x \right ) b}d x \right )+c_{1} \right ) {\mathrm e}^{-\left (\int \frac {\sin \left (x \right )}{x \sin \left (x \right )-\cos \left (x \right ) b}d x \right )} \\ \end{align*} |
Problem 1081
| ODE |
\[ \boxed {x^{4} y^{4}+x^{5} y^{3} y^{\prime }=0} \] |
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program solution |
\[ y = \frac {{\mathrm e}^{-c_{1}}}{x} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= \frac {c_{1}}{x} \\ \end{align*} |
Problem 1082
| ODE |
\[ \boxed {y \left (x \cos \left (x \right )+2 \sin \left (x \right )\right )+x \left (1+y\right ) y^{\prime }=0} \] |
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program solution |
\[ y = {\mathrm e}^{-\operatorname {LambertW}\left ({\mathrm e}^{-\sin \left (x \right )-2 \,\operatorname {Si}\left (x \right )-c_{1}}\right )-\sin \left (x \right )-2 \,\operatorname {Si}\left (x \right )-c_{1}} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \operatorname {LambertW}\left ({\mathrm e}^{-\sin \left (x \right )-2 \,\operatorname {Si}\left (x \right )-c_{1}}\right ) \] |
Problem 1083
| ODE |
\[ \boxed {x^{4} y^{3}+y+\left (y^{2} x^{5}-x \right ) y^{\prime }=0} \] |
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program solution |
\[ \frac {y \left (x^{4} y^{2}-3\right )}{3 x} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {\left (4 x^{3}+4 \sqrt {x^{6}-4 c_{1}^{3}}\right )^{\frac {2}{3}}+4 c_{1}}{2 x^{2} \left (4 x^{3}+4 \sqrt {x^{6}-4 c_{1}^{3}}\right )^{\frac {1}{3}} \sqrt {c_{1}}} \\ y \left (x \right ) &= \frac {-i \left (4 x^{3}+4 \sqrt {x^{6}-4 c_{1}^{3}}\right )^{\frac {2}{3}} \sqrt {3}+4 i \sqrt {3}\, c_{1} -\left (4 x^{3}+4 \sqrt {x^{6}-4 c_{1}^{3}}\right )^{\frac {2}{3}}-4 c_{1}}{4 x^{2} \left (4 x^{3}+4 \sqrt {x^{6}-4 c_{1}^{3}}\right )^{\frac {1}{3}} \sqrt {c_{1}}} \\ y \left (x \right ) &= -\frac {-i \left (4 x^{3}+4 \sqrt {x^{6}-4 c_{1}^{3}}\right )^{\frac {2}{3}} \sqrt {3}+4 i \sqrt {3}\, c_{1} +\left (4 x^{3}+4 \sqrt {x^{6}-4 c_{1}^{3}}\right )^{\frac {2}{3}}+4 c_{1}}{4 x^{2} \left (4 x^{3}+4 \sqrt {x^{6}-4 c_{1}^{3}}\right )^{\frac {1}{3}} \sqrt {c_{1}}} \\ \end{align*} |
Problem 1084
| ODE |
\[ \boxed {3 y x +2 y^{2}+y+\left (x^{2}+2 y x +x +2 y\right ) y^{\prime }=0} \] |
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program solution |
\[ x y \left (x^{2}+\left (2+y\right ) x +2 y+1\right )+y^{2} = c_{1} \] Verified OK.
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Maple solution | \begin{align*} y \left (x \right ) &= \frac {-\sqrt {4+x^{2} \left (x +1\right )^{2} c_{1}^{2}}+\left (-x^{2}-x \right ) c_{1}}{2 c_{1} \left (x +1\right )} \\ y \left (x \right ) &= \frac {\sqrt {4+x^{2} \left (x +1\right )^{2} c_{1}^{2}}+\left (-x^{2}-x \right ) c_{1}}{2 c_{1} \left (x +1\right )} \\ \end{align*} |
Problem 1085
| ODE |
\[ \boxed {12 y x +6 y^{3}+\left (9 x^{2}+10 x y^{2}\right ) y^{\prime }=0} \] |
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program solution |
\[ y^{3} x^{3} \left (2 y^{2}+3 x \right ) = c_{1} \] Verified OK.
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Maple solution | \[ \ln \left (x \right )-c_{1} +\frac {6 \ln \left (\frac {y \left (x \right )}{\sqrt {x}}\right )}{11}+\frac {2 \ln \left (\frac {2 y \left (x \right )^{2}+3 x}{x}\right )}{11} = 0 \] |
Problem 1086
| ODE |
\[ \boxed {3 x^{2} y^{2}+2 y+2 y^{\prime } x=0} \] |
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program solution |
\[ y = \frac {2 c_{3}}{3 x \left (c_{3} x +1\right )} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {2}{\left (3 x +2 c_{1} \right ) x} \] |
Problem 1087
| ODE |
\[ \boxed {y^{\prime \prime }-7 y^{\prime }+10 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ y = {\mathrm e}^{5 x}-2 \,{\mathrm e}^{2 x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{5 x}-2 \,{\mathrm e}^{2 x} \] |
Problem 1088
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }+2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3, y^{\prime }\left (0\right ) = -2] \end {align*}
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program solution |
\[ y = {\mathrm e}^{x} \left (3 \cos \left (x \right )-5 \sin \left (x \right )\right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{x} \left (-5 \sin \left (x \right )+3 \cos \left (x \right )\right ) \] |
Problem 1089
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }+2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = k_{0}, y^{\prime }\left (0\right ) = k_{1}] \end {align*}
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program solution |
\[ y = \left (\left (k_{1} -k_{0} \right ) \sin \left (x \right )+k_{0} \cos \left (x \right )\right ) {\mathrm e}^{x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{x} \left (\left (k_{1} -k_{0} \right ) \sin \left (x \right )+k_{0} \cos \left (x \right )\right ) \] |
Problem 1090
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 7, y^{\prime }\left (0\right ) = 4] \end {align*}
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program solution |
\[ y = {\mathrm e}^{x} \left (7-3 x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{x} \left (7-3 x \right ) \] |
Problem 1091
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = k_{0}, y^{\prime }\left (0\right ) = k_{1}] \end {align*}
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program solution |
\[ y = {\mathrm e}^{x} \left (-k_{0} x +k_{1} x +k_{0} \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = -{\mathrm e}^{x} \left (\left (x -1\right ) k_{0} -x k_{1} \right ) \] |
Problem 1092
| ODE |
\[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+4 y^{\prime } x +2 y=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -5, y^{\prime }\left (0\right ) = 1] \end {align*}
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program solution |
\[ y = -\frac {x -5}{x^{2}-1} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {-x +5}{x^{2}-1} \] |
Problem 1093
| ODE |
\[ \boxed {y^{\prime \prime }-2 y^{\prime }-3 y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-x}+\frac {c_{2} {\mathrm e}^{3 x}}{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{-x} c_{1} +c_{2} {\mathrm e}^{3 x} \] |
Problem 1094
| ODE |
\[ \boxed {y^{\prime \prime }-6 y^{\prime }+9 y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{3 x}+c_{2} x \,{\mathrm e}^{3 x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{3 x} \left (c_{2} x +c_{1} \right ) \] |
Problem 1095
| ODE |
\[ \boxed {y^{\prime \prime }-2 a y^{\prime }+a^{2} y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{a x}+c_{2} x \,{\mathrm e}^{a x} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = {\mathrm e}^{a x} \left (c_{2} x +c_{1} \right ) \] |
Problem 1096
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -y=0} \] |
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program solution |
\[ y = -\frac {c_{1}}{2 x}+c_{2} x \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \frac {c_{1} x^{2}+c_{2}}{x} \] |
Problem 1097
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-y^{\prime } x +y=0} \] |
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program solution |
\[ y = c_{1} x +\ln \left (x \right ) c_{2} x \] Verified OK.
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Maple solution | \[ y \left (x \right ) = x \left (c_{2} \ln \left (x \right )+c_{1} \right ) \] |
Problem 1098
| ODE |
\[ \boxed {x^{2} y^{\prime \prime }-\left (2 a -1\right ) x y^{\prime }+a^{2} y=0} \] |
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program solution |
\[ y = c_{1} x^{a}+c_{2} x^{a} \ln \left (x \right ) \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \left (c_{2} \ln \left (x \right )+c_{1} \right ) x^{a} \] |
Problem 1099
| ODE |
\[ \boxed {4 x^{2} y^{\prime \prime }-4 y^{\prime } x +\left (-16 x^{2}+3\right ) y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{-2 x} \sqrt {x}+\frac {c_{2} {\mathrm e}^{2 x} \sqrt {x}}{4} \] Verified OK.
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Maple solution | \[ y \left (x \right ) = \sqrt {x}\, \left (c_{1} \sinh \left (2 x \right )+c_{2} \cosh \left (2 x \right )\right ) \] |
Problem 1100
| ODE |
\[ \boxed {\left (x -1\right ) y^{\prime \prime }-y^{\prime } x +y=0} \] |
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program solution |
\[ y = c_{1} {\mathrm e}^{x}-c_{2} x \] Verified OK.
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Maple solution | \[ y \left (x \right ) = c_{1} x +c_{2} {\mathrm e}^{x} \] |