2.17.59 Problems 5801 to 5900

Problem 5801

ODE	\[ \boxed {2 y+\left (y x^{2}+1\right ) x y^{\prime }=0} \]
program solution	\[ -\frac {1}{y x^{2}}+\ln \left (y x^{2}\right )-2 \ln \left (x \right )-c_{1} = 0 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1}{\operatorname {LambertW}\left (\frac {c_{1}}{x^{2}}\right ) x^{2}} \]

Problem 5802

ODE	\[ \boxed {y \left (1+y x \right )+\left (1-y x \right ) x y^{\prime }=0} \]
program solution	\[ \ln \left (y x \right )+\frac {1}{y x}-2 \ln \left (x \right )-c_{1} = 0 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {1}{\operatorname {LambertW}\left (-\frac {c_{1}}{x^{2}}\right ) x} \]

Problem 5803

ODE	\[ \boxed {y \left (x^{2} y^{2}+1\right )+\left (x^{2} y^{2}-1\right ) x y^{\prime }=0} \]
program solution	\[ \frac {x^{2} y^{2}}{2}-\ln \left (y x \right )+2 \ln \left (x \right )-c_{1} = 0 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-2 c_{1}} x}{\sqrt {-\frac {x^{4} {\mathrm e}^{-4 c_{1}}}{\operatorname {LambertW}\left (-x^{4} {\mathrm e}^{-4 c_{1}}\right )}}} \]

Problem 5804

ODE	\[ \boxed {\left (x^{2}-y^{4}\right ) y^{\prime }-y x=0} \]
program solution	\[ \frac {\sqrt {\frac {y^{4}}{x^{2}}+1}\, \sqrt {x}}{y} = \frac {c_{2}}{\sqrt {x}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {-2 \sqrt {c_{1}^{2}-4 x^{2}}+2 c_{1}}}{2} \\ y \left (x \right ) &= \frac {\sqrt {-2 \sqrt {c_{1}^{2}-4 x^{2}}+2 c_{1}}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {2 \sqrt {c_{1}^{2}-4 x^{2}}+2 c_{1}}}{2} \\ y \left (x \right ) &= \frac {\sqrt {2 \sqrt {c_{1}^{2}-4 x^{2}}+2 c_{1}}}{2} \\ \end{align*}

Problem 5805

ODE	\[ \boxed {y \left (1+\sqrt {y^{4} x^{2}-1}\right )+2 y^{\prime } x=0} \]
program solution	\[ -\frac {\arctan \left (\frac {1}{\sqrt {y^{4} x^{2}-1}}\right )}{2}+\frac {\ln \left (x \right )}{2}-c_{1} = 0 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (-\ln \left (x \right )+c_{1} -2 \left (\int _{}^{\textit {\_Z}}\frac {1}{\textit {\_a} \sqrt {\textit {\_a}^{4}-1}}d \textit {\_a} \right )\right )}{\sqrt {x}} \]

Problem 5806

ODE	\[ \boxed {x \left (2-9 y^{2} x \right )+y \left (4 y^{2}-6 x^{3}\right ) y^{\prime }=0} \]
program solution	\[ -3 x^{3} y^{2}+y^{4}+x^{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {6 x^{3}-2 \sqrt {9 x^{6}-4 x^{2}-4 c_{1}}}}{2} \\ y \left (x \right ) &= \frac {\sqrt {6 x^{3}-2 \sqrt {9 x^{6}-4 x^{2}-4 c_{1}}}}{2} \\ y \left (x \right ) &= -\frac {\sqrt {6 x^{3}+2 \sqrt {9 x^{6}-4 x^{2}-4 c_{1}}}}{2} \\ y \left (x \right ) &= \frac {\sqrt {6 x^{3}+2 \sqrt {9 x^{6}-4 x^{2}-4 c_{1}}}}{2} \\ \end{align*}

Problem 5807

ODE	\[ \boxed {\frac {y}{x}+\left (y^{3}+\ln \left (x \right )\right ) y^{\prime }=0} \]
program solution	\[ y \ln \left (x \right )+\frac {y^{4}}{4} = c_{1} \] Verified OK.
Maple solution	\[ \ln \left (x \right ) y \left (x \right )+\frac {y \left (x \right )^{4}}{4}+c_{1} = 0 \]

Problem 5808

ODE	\[ \boxed {y^{\prime } \left (2 y-2\right )=-2 x -3} \]
program solution	\[ -y^{2}-x^{2}+2 y-3 x = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 1-\sqrt {-x^{2}-c_{1} -3 x +1} \\ y \left (x \right ) &= 1+\sqrt {-x^{2}-c_{1} -3 x +1} \\ \end{align*}

Problem 5809

ODE	\[ \boxed {4 y+\left (2 x -2 y\right ) y^{\prime }=-2 x} \]
program solution	\[ \frac {\ln \left (y^{2}-3 y x -x^{2}\right )}{2}+\frac {\sqrt {13}\, \operatorname {arctanh}\left (\frac {\left (-2 y+3 x \right ) \sqrt {13}}{13 x}\right )}{13} = c_{1} \] Verified OK.
Maple solution	\[ -\frac {\ln \left (\frac {-x^{2}-3 x y \left (x \right )+y \left (x \right )^{2}}{x^{2}}\right )}{2}+\frac {\sqrt {13}\, \operatorname {arctanh}\left (\frac {\left (2 y \left (x \right )-3 x \right ) \sqrt {13}}{13 x}\right )}{13}-\ln \left (x \right )-c_{1} = 0 \]

Problem 5810

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }-y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\left (1+\sqrt {2}\right ) x}+\frac {c_{2} \sqrt {2}\, {\mathrm e}^{\left (\sqrt {2}-1\right ) x}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{\left (\sqrt {2}-1\right ) x}+c_{2} {\mathrm e}^{-\left (1+\sqrt {2}\right ) x} \]

Problem 5811

ODE	\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}-\frac {y}{x^{2}}=0} \]
program solution	\[ y = -\frac {c_{1}}{2 x}+c_{2} x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{2}+c_{1}}{x} \]

Problem 5812

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y^{\prime } x +y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{i \operatorname {arcsinh}\left (x \right )}+\frac {i c_{2}}{2 \,{\mathrm e}^{i \operatorname {arcsinh}\left (x \right )}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (\operatorname {arcsinh}\left (x \right )\right )+c_{2} \cos \left (\operatorname {arcsinh}\left (x \right )\right ) \]

Problem 5813

ODE	\[ \boxed {y^{\prime \prime }-\cot \left (x \right ) y^{\prime }+y \cos \left (x \right )=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \left (1+\cos \left (x \right )\right ) \operatorname {HeunC}\left (0, 1, -1, -2, \frac {3}{2}, \frac {\cos \left (x \right )}{2}+\frac {1}{2}\right ) \left (c_{1} +c_{2} \left (\int _{}^{\cos \left (x \right )}\frac {1}{\left (\textit {\_a} +1\right )^{2} \operatorname {HeunC}\left (0, 1, -1, -2, \frac {3}{2}, \frac {\textit {\_a}}{2}+\frac {1}{2}\right )^{2}}d \textit {\_a} \right )\right ) \]

Problem 5814

ODE	\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}+y x^{2}=0} \]
program solution	\[ y = c_{1} \operatorname {BesselJ}\left (0, \frac {x^{2}}{2}\right )+c_{2} \operatorname {BesselY}\left (0, \frac {x^{2}}{2}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {BesselJ}\left (0, \frac {x^{2}}{2}\right )+c_{2} \operatorname {BesselY}\left (0, \frac {x^{2}}{2}\right ) \]

Problem 5815

ODE	\[ \boxed {x^{2} \left (-x^{2}+1\right ) y^{\prime \prime }+2 x \left (-x^{2}+1\right ) y^{\prime }-2 y=0} \]
program solution	\[ y = c_{3} \left (-c_{1} \ln \left (1+x \right )+c_{1} \ln \left (x -1\right )+4 c_{2} +\frac {c_{1} \ln \left (1+x \right )}{x^{2}}-\frac {c_{1} \ln \left (x -1\right )}{x^{2}}-\frac {2 c_{1}}{x}-\frac {4 c_{2}}{x^{2}}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} \left (x^{2}-1\right ) \ln \left (x -1\right )+\left (-x^{2}+1\right ) c_{2} \ln \left (x +1\right )+2 c_{1} x^{2}-2 c_{2} x -2 c_{1}}{2 x^{2}} \]

Problem 5816

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +y=0} \]
program solution	\[ y = c_{1} x +c_{2} \sqrt {x -1}\, \sqrt {1+x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x +c_{2} \sqrt {x -1}\, \sqrt {x +1} \]

Problem 5817

ODE	\[ \boxed {y^{\prime \prime \prime }-2 x y^{\prime \prime }+4 x^{2} y^{\prime }+8 y x^{3}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 5818

ODE	\[ \boxed {y^{\prime \prime }+x \left (1-x \right ) y^{\prime }+{\mathrm e}^{x} y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 5819

ODE	\[ \boxed {x^{2} y^{\prime \prime }+2 y^{\prime } x +4 y=0} \]
program solution	\[ y = c_{1} x^{-\frac {1}{2}-\frac {i \sqrt {15}}{2}}-\frac {i c_{2} \sqrt {15}\, x^{-\frac {1}{2}+\frac {i \sqrt {15}}{2}}}{15} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sin \left (\frac {\sqrt {15}\, \ln \left (x \right )}{2}\right )+c_{2} \cos \left (\frac {\sqrt {15}\, \ln \left (x \right )}{2}\right )}{\sqrt {x}} \]

Problem 5820

ODE	\[ \boxed {x^{4} y^{\prime \prime \prime \prime }-x^{2} y^{\prime \prime }+y=0} \]
program solution	\[ \text {Expression too large to display} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \moverset {4}{\munderset {\textit {\_a} =1}{\sum }}x^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-6 \textit {\_Z}^{3}+10 \textit {\_Z}^{2}-5 \textit {\_Z} +1, \operatorname {index} =\textit {\_a} \right )} \textit {\_C}_{\textit {\_a}} \]

Problem 5821

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y^{\prime } x +y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{i \operatorname {arcsinh}\left (x \right )}+\frac {i c_{2}}{2 \,{\mathrm e}^{i \operatorname {arcsinh}\left (x \right )}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sin \left (\operatorname {arcsinh}\left (x \right )\right )+c_{2} \cos \left (\operatorname {arcsinh}\left (x \right )\right ) \]

Problem 5822

ODE	\[ \boxed {y^{\prime \prime }+y^{\prime } x +y=2 x \,{\mathrm e}^{x}-1} \]
program solution	\[ y = 2 i \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{-\frac {1}{2}-\frac {x^{2}}{2}} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (1+x \right )}{2}\right )+\frac {\left (-i c_{1} \sqrt {\pi }\, \sqrt {2}\, \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right )+2 c_{2} \right ) {\mathrm e}^{-\frac {x^{2}}{2}}}{2}+2 \,{\mathrm e}^{x}-1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2 i \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{-\frac {x^{2}}{2}-\frac {1}{2}} \operatorname {erf}\left (\frac {i \sqrt {2}\, \left (x +1\right )}{2}\right )+\left (c_{1} \operatorname {erf}\left (\frac {i \sqrt {2}\, x}{2}\right )+c_{2} \right ) {\mathrm e}^{-\frac {x^{2}}{2}}+2 \,{\mathrm e}^{x}-1 \]

Problem 5823

ODE	\[ \boxed {x y^{\prime \prime }+y^{\prime } x -y=x^{2}+2 x} \]
program solution	\[ y = -c_{2} {\mathrm e}^{-x}+x \left (\operatorname {expIntegral}_{1}\left (x \right ) c_{2} +c_{1} \right )+x \left (\operatorname {expIntegral}_{1}\left (x \right ) x^{2} {\mathrm e}^{x}-\left (\int _{0}^{x}\left (\alpha +2\right ) \left (\operatorname {expIntegral}_{1}\left (\alpha \right ) \alpha \,{\mathrm e}^{\alpha }-1\right )d \alpha \right )-x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -c_{2} {\mathrm e}^{-x}+x \left (c_{2} \operatorname {expIntegral}_{1}\left (x \right )+x +c_{1} \right ) \]

Problem 5824

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } x -y=x^{2}+2 x} \]
program solution	\[ y = x \left (\frac {x}{3}-\frac {c_{1}}{2 x^{2}}+\ln \left (x \right )+c_{2} \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{x}+c_{2} x +\frac {\left (x +3 \ln \left (x \right )\right ) x}{3} \]

Problem 5825

ODE	\[ \boxed {x^{3} y^{\prime \prime }+y^{\prime } x -y=\cos \left (\frac {1}{x}\right )} \]
program solution	\[ y = x \left (c_{1} {\mathrm e}^{\frac {1}{x}}+c_{2} \right )-\frac {x \left (\sin \left (\frac {1}{x}\right )+\cos \left (\frac {1}{x}\right )\right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x \left (-2 \,{\mathrm e}^{\frac {1}{x}} c_{2} +\cos \left (\frac {1}{x}\right )+\sin \left (\frac {1}{x}\right )-2 c_{1} \right )}{2} \]

Problem 5826

ODE	\[ \boxed {x \left (1+x \right ) y^{\prime \prime }+\left (x +2\right ) y^{\prime }-y=x +\frac {1}{x}} \]
program solution	\[ y = \frac {2 \ln \left (x \right ) x^{2}+4 c_{2} x^{2}+4 \ln \left (x \right ) x +8 c_{2} x -2 c_{1} +4 c_{2} +6 x +5}{4 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2 \ln \left (x \right ) x^{2}+4 c_{2} x^{2}+4 \ln \left (x \right ) x +8 c_{2} x +4 c_{1} +4 c_{2} +6 x +5}{4 x} \]

Problem 5827

ODE	\[ \boxed {2 x y^{\prime \prime }+\left (x -2\right ) y^{\prime }-y=x^{2}-1} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {x}{2}}+2 c_{2} \left (x -2\right )+x^{2}-4 x +9 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-2+x \right ) c_{2} +c_{1} {\mathrm e}^{-\frac {x}{2}}+x^{2}+1 \]

Problem 5828

ODE	\[ \boxed {x^{2} \left (1+x \right ) y^{\prime \prime }+x \left (4 x +3\right ) y^{\prime }-y=x +\frac {1}{x}} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {-5 x^{-\sqrt {2}} \left (\sqrt {2}-\frac {6}{5}\right ) \operatorname {hypergeom}\left (\left [2-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right ) \left (\int \frac {x^{\sqrt {2}-1} \left (\left (\left (-2 x^{2}-4 x -\frac {5}{2}\right ) \sqrt {2}-x^{2}-\frac {11 x}{2}-3\right ) \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right )+\left (\sqrt {2}-1\right ) x \left (x^{2}+\frac {1}{2} x +\frac {1}{2}\right ) \operatorname {hypergeom}\left (\left [\sqrt {2}, \sqrt {2}\right ], \left [2+2 \sqrt {2}\right ], -x \right )\right ) \left (x^{2}+1\right )}{\left (-7 \sqrt {2}\, \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right )+4 x \operatorname {hypergeom}\left (\left [\sqrt {2}, \sqrt {2}\right ], \left [2+2 \sqrt {2}\right ], -x \right ) \left (\sqrt {2}-\frac {11}{8}\right )\right ) \operatorname {hypergeom}\left (\left [-1-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right )+4 x \operatorname {hypergeom}\left (\left [-\sqrt {2}, -\sqrt {2}\right ], \left [2-2 \sqrt {2}\right ], -x \right ) \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right ) \left (\sqrt {2}+\frac {11}{8}\right )}d x \right )+5 x^{\sqrt {2}} \left (\sqrt {2}+\frac {6}{5}\right ) \operatorname {hypergeom}\left (\left [\sqrt {2}-1, 2+\sqrt {2}\right ], \left [1+2 \sqrt {2}\right ], -x \right ) \left (\int \frac {x^{-1-\sqrt {2}} \left (x^{2}+1\right ) \left (\left (\left (-2 x^{2}-4 x -\frac {5}{2}\right ) \sqrt {2}+x^{2}+\frac {11 x}{2}+3\right ) \operatorname {hypergeom}\left (\left [-1-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right )+x \operatorname {hypergeom}\left (\left [-\sqrt {2}, -\sqrt {2}\right ], \left [2-2 \sqrt {2}\right ], -x \right ) \left (x^{2}+\frac {1}{2} x +\frac {1}{2}\right ) \left (1+\sqrt {2}\right )\right )}{\left (-7 \sqrt {2}\, \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right )+4 x \operatorname {hypergeom}\left (\left [\sqrt {2}, \sqrt {2}\right ], \left [2+2 \sqrt {2}\right ], -x \right ) \left (\sqrt {2}-\frac {11}{8}\right )\right ) \operatorname {hypergeom}\left (\left [-1-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right )+4 x \operatorname {hypergeom}\left (\left [-\sqrt {2}, -\sqrt {2}\right ], \left [2-2 \sqrt {2}\right ], -x \right ) \operatorname {hypergeom}\left (\left [\sqrt {2}-1, \sqrt {2}-1\right ], \left [1+2 \sqrt {2}\right ], -x \right ) \left (\sqrt {2}+\frac {11}{8}\right )}d x \right )+2 x^{-\sqrt {2}} \operatorname {hypergeom}\left (\left [2-\sqrt {2}, -1-\sqrt {2}\right ], \left [1-2 \sqrt {2}\right ], -x \right ) c_{2} +2 x^{\sqrt {2}} \operatorname {hypergeom}\left (\left [\sqrt {2}-1, 2+\sqrt {2}\right ], \left [1+2 \sqrt {2}\right ], -x \right ) c_{1}}{2 x} \]

Problem 5829

ODE	\[ \boxed {x^{2} \left (-1+\ln \left (x \right )\right ) y^{\prime \prime }-y^{\prime } x +y=x \left (-\ln \left (x \right )+1\right )^{2}} \]
program solution	\[ y = \frac {\ln \left (x \right )^{2} x}{2}+\left (-x +c_{1} \right ) \ln \left (x \right )-c_{2} x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\ln \left (x \right )^{2} x}{2}+\left (-x -c_{1} \right ) \ln \left (x \right )+c_{2} x \]

Problem 5830

ODE	\[ \boxed {x y^{\prime \prime }+2 y^{\prime }+y x=\sec \left (x \right )} \]
program solution	\[ y = \frac {c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )}{x}+\frac {\ln \left (\cos \left (x \right )\right ) \cos \left (x \right )}{x}+\sin \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-\ln \left (\sec \left (x \right )\right ) \cos \left (x \right )+\cos \left (x \right ) c_{1} +\sin \left (x \right ) \left (x +c_{2} \right )}{x} \]

Problem 5831

ODE	\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +\frac {y}{4}=-\frac {x^{2}}{2}+\frac {1}{2}} \]
program solution	\[ y = c_{1} \sqrt {1+x}+c_{2} \sqrt {x -1}+\frac {2 x^{2}}{15}+\frac {14}{15} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2 \left (x^{2}+7\right ) \sqrt {x +\sqrt {x^{2}-1}}+15 c_{1} x +15 c_{1} \sqrt {x^{2}-1}+15 c_{2}}{15 \sqrt {x +\sqrt {x^{2}-1}}} \]

Problem 5832

ODE	\[ \boxed {\left (\cos \left (x \right )+\sin \left (x \right )\right ) y^{\prime \prime }-2 \cos \left (x \right ) y^{\prime }+\left (\cos \left (x \right )-\sin \left (x \right )\right ) y=\left (\cos \left (x \right )+\sin \left (x \right )\right )^{2} {\mathrm e}^{2 x}} \]
program solution	\[ y = -2 c_{1} {\mathrm e}^{x}-2 c_{2} \cos \left (x \right )+\left (\int _{0}^{x}\frac {{\mathrm e}^{\alpha } \cos \left (\alpha \right )^{2} \left (\cos \left (\alpha \right )+\sin \left (\alpha \right )\right )}{\sin \left (\alpha \right )+\left (\cos \left (\alpha \right )+\sin \left (\alpha \right )\right ) \cos \left (\alpha \right )}d \alpha \right ) {\mathrm e}^{x}-\left (\int _{0}^{x}\frac {\cos \left (\alpha \right ) \left (\cos \left (\alpha \right )+\sin \left (\alpha \right )\right ) {\mathrm e}^{2 \alpha }}{\sin \left (\alpha \right )+\left (\cos \left (\alpha \right )+\sin \left (\alpha \right )\right ) \cos \left (\alpha \right )}d \alpha \right ) \cos \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\cos \left (x \right ) \left (\left (\int {\mathrm e}^{\int \frac {\left (-\cot \left (x \right )+1\right ) \cos \left (x \right )+2 \sin \left (x \right ) \left (\tan \left (x \right )+1\right )}{\cos \left (x \right )+\sin \left (x \right )}d x} \sin \left (x \right )d x \right ) c_{1} -\left (\int {\mathrm e}^{2 x -2 \left (\int \frac {\sin \left (x \right )}{\cos \left (x \right )+\sin \left (x \right )}d x \right )-2 \left (\int \frac {\sin \left (x \right ) \tan \left (x \right )}{\cos \left (x \right )+\sin \left (x \right )}d x \right )+\int \frac {\cos \left (x \right ) \cot \left (x \right )}{\cos \left (x \right )+\sin \left (x \right )}d x -\left (\int \frac {\cos \left (x \right )}{\cos \left (x \right )+\sin \left (x \right )}d x \right )} \left (\csc \left (x \right )+\sec \left (x \right )\right )d x \right ) \left (\int {\mathrm e}^{2 \left (\int \frac {\sin \left (x \right )}{\cos \left (x \right )+\sin \left (x \right )}d x \right )+2 \left (\int \frac {\sin \left (x \right ) \tan \left (x \right )}{\cos \left (x \right )+\sin \left (x \right )}d x \right )-\left (\int \frac {\cos \left (x \right ) \cot \left (x \right )}{\cos \left (x \right )+\sin \left (x \right )}d x \right )+\int \frac {\cos \left (x \right )}{\cos \left (x \right )+\sin \left (x \right )}d x} \sin \left (x \right )d x \right )+\int {\mathrm e}^{2 x -2 \left (\int \frac {\sin \left (x \right )}{\cos \left (x \right )+\sin \left (x \right )}d x \right )-2 \left (\int \frac {\sin \left (x \right ) \tan \left (x \right )}{\cos \left (x \right )+\sin \left (x \right )}d x \right )+\int \frac {\cos \left (x \right ) \cot \left (x \right )}{\cos \left (x \right )+\sin \left (x \right )}d x -\left (\int \frac {\cos \left (x \right )}{\cos \left (x \right )+\sin \left (x \right )}d x \right )} \left (\csc \left (x \right )+\sec \left (x \right )\right ) \left (\int {\mathrm e}^{2 \left (\int \frac {\sin \left (x \right )}{\cos \left (x \right )+\sin \left (x \right )}d x \right )+2 \left (\int \frac {\sin \left (x \right ) \tan \left (x \right )}{\cos \left (x \right )+\sin \left (x \right )}d x \right )-\left (\int \frac {\cos \left (x \right ) \cot \left (x \right )}{\cos \left (x \right )+\sin \left (x \right )}d x \right )+\int \frac {\cos \left (x \right )}{\cos \left (x \right )+\sin \left (x \right )}d x} \sin \left (x \right )d x \right )d x -c_{2} \right ) \]

Problem 5833

ODE	\[ \boxed {\left (\cos \left (x \right )-\sin \left (x \right )\right ) y^{\prime \prime }-2 \sin \left (x \right ) y^{\prime }+\left (\cos \left (x \right )+\sin \left (x \right )\right ) y=\left (\cos \left (x \right )-\sin \left (x \right )\right )^{2}} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 5834

ODE	\[ \boxed {y^{\prime }-x^{2} \left (1+y^{2}\right )=0} \]
program solution	\[ y = \frac {-c_{3} \cos \left (\frac {x^{3}}{3}\right )+\sin \left (\frac {x^{3}}{3}\right )}{c_{3} \sin \left (\frac {x^{3}}{3}\right )+\cos \left (\frac {x^{3}}{3}\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \tan \left (\frac {x^{3}}{3}+c_{1} \right ) \]

Problem 5835

ODE	\[ \boxed {y^{\prime }-\frac {x^{2}}{1-y^{2}}=0} \]
program solution	\[ -\frac {x^{3}}{3}-\frac {y^{3}}{3}+y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{\frac {2}{3}}+4}{2 \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{\frac {2}{3}}-4 i \sqrt {3}+4}{4 \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{\frac {2}{3}} \sqrt {3}-4 i \sqrt {3}-\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{\frac {2}{3}}-4}{4 \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 5836

ODE	\[ \boxed {y^{\prime }-\frac {3 x^{2}+4 x +2}{2 y-2}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = -1] \end {align*}
program solution	\[ -x^{3}-2 x^{2}+y^{2}-2 x -2 y = 3 \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1-\sqrt {\left (x +2\right ) \left (x^{2}+2\right )} \]

Problem 5837

ODE	\[ \boxed {y^{\prime } x -2 \sqrt {y x}-y=0} \]
program solution	\[ y = \ln \left (x \right )^{2} x -2 x \ln \left (x \right ) c_{1} +c_{1}^{2} x \] Verified OK.
Maple solution	\[ -\frac {y \left (x \right )}{\sqrt {x y \left (x \right )}}+\ln \left (x \right )-c_{1} = 0 \]

Problem 5838

ODE	\[ \boxed {y^{\prime }-\frac {x +y-1}{x -y+3}=0} \]
program solution	\[ \frac {\ln \left (y^{2}+x^{2}-4 y+2 x +5\right )}{2}-\arctan \left (\frac {y-2}{1+x}\right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2+\tan \left (\operatorname {RootOf}\left (2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (x +1\right )+2 c_{1} \right )\right ) \left (-x -1\right ) \]

Problem 5839

ODE	\[ \boxed {y+\left (x -2 \sin \left (y\right )\right ) y^{\prime }=-{\mathrm e}^{x}} \]
program solution	\[ y x +{\mathrm e}^{x}+2 \cos \left (y\right ) = c_{1} \] Verified OK.
Maple solution	\[ {\mathrm e}^{x}+x y \left (x \right )+2 \cos \left (y \left (x \right )\right )+c_{1} = 0 \]

Problem 5840

ODE	\[ \boxed {\frac {6}{y}+\left (\frac {x^{2}}{y}+\frac {3 y}{x}\right ) y^{\prime }=-3 x} \]
program solution	\[ y x^{3}+y^{3}+3 x^{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-12 x^{3}+\left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {2}{3}}}{6 \left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}}{12}-\frac {\left (i \sqrt {3}-1\right ) x^{3}}{\left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {12 i \sqrt {3}\, x^{3}+i \left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+12 x^{3}-\left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {2}{3}}}{12 \left (-324 x^{2}-108 c_{1} +12 \sqrt {12 x^{9}+729 x^{4}+486 c_{1} x^{2}+81 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 5841

ODE	\[ \boxed {y^{2}-y x +x^{2} y^{\prime }=0} \]
program solution	\[ y = \frac {x}{\ln \left (x \right )+c_{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x}{\ln \left (x \right )+c_{1}} \]

Problem 5842

ODE	\[ \boxed {y-\left (x -y\right ) y^{\prime }=-x} \]
program solution	\[ \frac {\ln \left (x^{2}+y^{2}\right )}{2}+\arctan \left (\frac {x}{y}\right ) = c_{1} \] Verified OK.
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 5843

ODE	\[ \boxed {y^{\prime }-\frac {y}{2 x}-\frac {x^{2}}{2 y}=0} \]
program solution	\[ -\frac {x^{2}}{2}+\frac {y^{2}}{x} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {2}\, \sqrt {x \left (x^{2}+2 c_{1} \right )}}{2} \\ y \left (x \right ) &= \frac {\sqrt {2}\, \sqrt {x \left (x^{2}+2 c_{1} \right )}}{2} \\ \end{align*}

Problem 5844

ODE	\[ \boxed {y^{\prime }-\frac {y}{t}-\frac {y^{2}}{t}=-\frac {2}{t}} \]
program solution	\[ y = \frac {-2 t^{3}+c_{3}}{t^{3}+c_{3}} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {-2 c_{1} t^{3}-1}{c_{1} t^{3}-1} \]

Problem 5845

ODE	\[ \boxed {y^{\prime }+\frac {y}{t}+y^{2}=-1} \]
program solution	\[ y = \frac {-c_{3} \operatorname {BesselJ}\left (1, t\right )-\operatorname {BesselY}\left (1, t\right )}{c_{3} \operatorname {BesselJ}\left (0, t\right )+\operatorname {BesselY}\left (0, t\right )} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {-i \operatorname {BesselK}\left (1, i t \right ) c_{1} -\operatorname {BesselJ}\left (1, t\right )}{\operatorname {BesselK}\left (0, i t \right ) c_{1} +\operatorname {BesselJ}\left (0, t\right )} \]

Problem 5846

ODE	\[ \boxed {y y^{\prime }-a {y^{\prime }}^{2}=-x} \]
program solution	\[ y = -i a -i x \] Verified OK. \[ y = i a +i x \] Verified OK. \[ x = \frac {\left (y+\sqrt {y^{2}+4 a x}\right ) \left (a \,\operatorname {arcsinh}\left (\frac {y+\sqrt {y^{2}+4 a x}}{2 a}\right )+c_{1} \right ) \sqrt {2}}{2 \sqrt {\frac {y \sqrt {y^{2}+4 a x}+2 a^{2}+2 a x +y^{2}}{a^{2}}}\, a} \] Verified OK. \[ x = \frac {\left (-y+\sqrt {y^{2}+4 a x}\right ) \left (a \,\operatorname {arcsinh}\left (\frac {-y+\sqrt {y^{2}+4 a x}}{2 a}\right )-c_{1} \right ) \sqrt {2}}{2 \sqrt {\frac {2 a^{2}+2 a x -y \sqrt {y^{2}+4 a x}+y^{2}}{a^{2}}}\, a} \] Verified OK.
Maple solution	\begin{align*} \frac {-\frac {\sqrt {2}\, \left (y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}\right ) \operatorname {arcsinh}\left (\frac {y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}}{2 a}\right )}{2}+x \sqrt {\frac {y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}+2 a^{2}+2 a x +y \left (x \right )^{2}}{a^{2}}}+c_{1} y \left (x \right )+c_{1} \sqrt {4 a x +y \left (x \right )^{2}}}{\sqrt {\frac {y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}+y \left (x \right )^{2}+2 a \left (a +x \right )}{a^{2}}}} &= 0 \\ \frac {\sqrt {\frac {-2 y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}+2 y \left (x \right )^{2}+4 a \left (a +x \right )}{a^{2}}}\, x -\left (y \left (x \right )-\sqrt {4 a x +y \left (x \right )^{2}}\right ) \left (-\operatorname {arcsinh}\left (\frac {-y \left (x \right )+\sqrt {4 a x +y \left (x \right )^{2}}}{2 a}\right )+c_{1} \right )}{\sqrt {\frac {-2 y \left (x \right ) \sqrt {4 a x +y \left (x \right )^{2}}+2 y \left (x \right )^{2}+4 a \left (a +x \right )}{a^{2}}}} &= 0 \\ \end{align*}

Problem 5847

ODE	\[ \boxed {{y^{\prime }}^{2}-a^{2} y^{2}=0} \]
program solution	\[ y = \frac {{\mathrm e}^{-a x}}{c_{2}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} {\mathrm e}^{a x} \\ y \left (x \right ) &= c_{1} {\mathrm e}^{-a x} \\ \end{align*}

Problem 5848

ODE	\[ \boxed {{y^{\prime }}^{2}=4 x^{2}} \]
program solution	\[ y = -x^{2}+c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= x^{2}+c_{1} \\ y \left (x \right ) &= -x^{2}+c_{1} \\ \end{align*}

Problem 5849

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }-3 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+\frac {c_{2} {\mathrm e}^{3 x}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{3 x} \]

Problem 5850

ODE	\[ \boxed {s^{\prime \prime }+2 s^{\prime }+s=0} \] With initial conditions \begin {align*} [s \left (0\right ) = 4, s^{\prime }\left (0\right ) = -2] \end {align*}
program solution	\[ s = 2 \left (t +2\right ) {\mathrm e}^{-t} \] Verified OK.
Maple solution	\[ s \left (t \right ) = 2 \,{\mathrm e}^{-t} \left (t +2\right ) \]

Problem 5851

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+5 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x} \cos \left (2 x \right )+\frac {c_{2} {\mathrm e}^{x} \sin \left (2 x \right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} \left (c_{1} \sin \left (2 x \right )+c_{2} \cos \left (2 x \right )\right ) \]

Problem 5852

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }-3 y=3 x +1} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+\frac {c_{2} {\mathrm e}^{3 x}}{4}-x +\frac {1}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{2} {\mathrm e}^{-x}+{\mathrm e}^{3 x} c_{1} -x +\frac {1}{3} \]

Problem 5853

ODE	\[ \boxed {y^{\prime \prime }-3 y^{\prime }+2 y={\mathrm e}^{2 x} x} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{2 x}+\frac {x^{2} {\mathrm e}^{2 x}}{2}-{\mathrm e}^{2 x} x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (x^{2}+2 c_{1} -2 x +2\right ) {\mathrm e}^{x}+2 c_{2} \right ) {\mathrm e}^{x}}{2} \]

Problem 5854

ODE	\[ \boxed {y^{\prime \prime }+y=4 \sin \left (x \right )} \]
program solution	\[ y = c_{1} \cos \left (x \right )+c_{2} \sin \left (x \right )-2 x \cos \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{1} -2 x \right ) \cos \left (x \right )+\sin \left (x \right ) \left (c_{2} +2\right ) \]

Problem 5855

ODE	\[ \boxed {y^{\prime \prime }+2 x^{2} y^{\prime }+\left (x^{4}+2 x -1\right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-\frac {x \left (x^{2}+3\right )}{3}}+\frac {c_{2} {\mathrm e}^{-\frac {x \left (x^{2}-3\right )}{3}}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {x \left (x^{2}-3\right )}{3}}+c_{2} {\mathrm e}^{-\frac {x \left (x^{2}+3\right )}{3}} \]

Problem 5856

ODE	\[ \boxed {p \,x^{2} u^{\prime \prime }+q x u^{\prime }+r u=f \left (x \right )} \]
program solution	\[ u = c_{1} x^{\frac {p -q +\sqrt {p^{2}-2 q p -4 p r +q^{2}}}{2 p}}-\frac {c_{2} p \,x^{-\frac {-p +q +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}}}{\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}+\frac {\left (\int _{0}^{x}\alpha ^{-\frac {3 p -q +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} f \left (\alpha \right )d \alpha \right ) x^{\frac {p -q +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}}-\left (\int _{0}^{x}\alpha ^{\frac {-3 p +q +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} f \left (\alpha \right )d \alpha \right ) x^{-\frac {-p +q +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}}}{\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}} \] Verified OK.
Maple solution	\[ u \left (x \right ) = \frac {x^{\frac {-q +p +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} c_{2} \sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}+x^{-\frac {q -p +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} c_{1} \sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}+x^{\frac {-q +p +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} \left (\int x^{-\frac {3 p -q +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} f \left (x \right )d x \right )-x^{-\frac {q -p +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} \left (\int x^{\frac {-3 p +q +\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}}{2 p}} f \left (x \right )d x \right )}{\sqrt {p^{2}+\left (-2 q -4 r \right ) p +q^{2}}} \]

Problem 5857

ODE	\[ \boxed {\sin \left (x \right ) u^{\prime \prime }+2 \cos \left (x \right ) u^{\prime }+\sin \left (x \right ) u=0} \]
program solution	\[ u = c_{1} \cos \left (\sqrt {2}\, x \right ) \csc \left (x \right )+\frac {c_{2} \sin \left (\sqrt {2}\, x \right ) \csc \left (x \right ) \sqrt {2}}{2} \] Verified OK.
Maple solution	\[ u \left (x \right ) = \csc \left (x \right ) \left (c_{1} \sin \left (\sqrt {2}\, x \right )+c_{2} \cos \left (\sqrt {2}\, x \right )\right ) \]

Problem 5858

ODE	Solve \begin {gather*} \boxed {3 \left (y^{\prime \prime }\right )^{2}-y^{\prime } y^{\prime \prime \prime }-y^{\prime \prime } \left (y^{\prime }\right )^{2}=0} \end {gather*}
program solution	N/A
Maple solution	\begin{align*} y \left (x \right ) &= c_{1} \\ y \left (x \right ) &= \frac {\operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {c_{3} +x}{c_{1}}}}{c_{2} c_{1}}\right ) c_{1} -c_{3} -x}{c_{1}} \\ \end{align*}

Problem 5859

ODE	\[ \boxed {y^{\prime \prime }-\frac {x y^{\prime }}{-x^{2}+1}+\frac {y}{-x^{2}+1}=0} \]
program solution	\[ y = -c_{1} x +c_{2} \sqrt {x -1}\, \sqrt {1+x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x +c_{2} \sqrt {x -1}\, \sqrt {x +1} \]

Problem 5860

ODE	\[ \boxed {x^{2} y y^{\prime \prime }-x^{2} {y^{\prime }}^{2}+y^{2}=0} \]
program solution
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= c_{2} x \,{\mathrm e}^{-c_{1} x +1} \\ \end{align*}

Problem 5861

ODE	\[ \boxed {y^{\prime \prime \prime }-3 y^{\prime \prime }+3 y^{\prime }-y=4 \,{\mathrm e}^{t}} \]
program solution	\[ y = {\mathrm e}^{t} \left (c_{3} t^{2}+c_{2} t +c_{1} \right )+\frac {2 \,{\mathrm e}^{t} t^{3}}{3} \] Verified OK.
Maple solution	\[ y \left (t \right ) = {\mathrm e}^{t} \left (\frac {2}{3} t^{3}+c_{1} +t c_{2} +t^{2} c_{3} \right ) \]

Problem 5862

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+2 y^{\prime \prime }+y=3 \sin \left (t \right )-5 \cos \left (t \right )} \]
program solution	\[ y = \left (c_{2} t +c_{1} \right ) {\mathrm e}^{i t}+\left (c_{4} t +c_{3} \right ) {\mathrm e}^{-i t}-\frac {\left (\int \left (-3 \sin \left (t \right )+5 \cos \left (t \right )\right ) \left (-\sin \left (t \right )+\cos \left (t \right ) t \right )d t \right ) \cos \left (t \right )}{2}-\frac {\left (\int \left (-3 \sin \left (t \right )+5 \cos \left (t \right )\right ) \left (\sin \left (t \right ) t +\cos \left (t \right )\right )d t \right ) \sin \left (t \right )}{2}-\frac {5 \left (\left (-\frac {3}{5}+i-4 t \right ) \cos \left (t \right )+\frac {3 \left (-5+i+4 t \right ) \sin \left (t \right )}{5}\right ) t}{16} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \frac {\left (5 t^{2}+\left (8 c_{3} -6\right ) t +8 c_{1} -10\right ) \cos \left (t \right )}{8}-\frac {3 \sin \left (t \right ) \left (t^{2}+\left (-\frac {8 c_{4}}{3}+\frac {10}{3}\right ) t -\frac {8 c_{2}}{3}-2\right )}{8} \]

Problem 5863

ODE	\[ \boxed {y^{\prime \prime \prime }-y^{\prime \prime }-y^{\prime }+y=g \left (t \right )} \]
program solution	\[ y = {\mathrm e}^{-t} c_{1} +{\mathrm e}^{t} \left (c_{3} t +c_{2} \right )+\frac {\left (\int g \left (t \right ) {\mathrm e}^{t}d t \right ) {\mathrm e}^{-t}}{4}-\frac {\left (\int g \left (t \right ) \left (2 t +1\right ) {\mathrm e}^{-t}d t \right ) {\mathrm e}^{t}}{4}+\frac {\left (\int g \left (t \right ) {\mathrm e}^{-t}d t \right ) {\mathrm e}^{t} t}{2} \] Verified OK.
Maple solution	\[ y \left (t \right ) = -\frac {\left (\int \left (2 t +1\right ) g \left (t \right ) {\mathrm e}^{-t}d t \right ) {\mathrm e}^{t}}{4}+\frac {\left (\int {\mathrm e}^{-t} g \left (t \right )d t \right ) {\mathrm e}^{t} t}{2}+\frac {\left (\int {\mathrm e}^{t} g \left (t \right )d t \right ) {\mathrm e}^{-t}}{4}+c_{2} {\mathrm e}^{-t}+{\mathrm e}^{t} \left (c_{3} t +c_{1} \right ) \]

Problem 5864

ODE	\[ \boxed {y^{\left (5\right )}-\frac {y^{\prime \prime \prime \prime }}{t}=0} \]
program solution	\[ y = \frac {c_{3}^{2} {\mathrm e}^{2 c_{2}} t^{5}}{60}-\frac {c_{1} t^{3}}{12}+\frac {c_{4} t^{2}}{2}+t c_{5} +c_{6} \] Verified OK.
Maple solution	\[ y \left (t \right ) = c_{3} t^{5}+c_{5} t^{3}+c_{2} t^{2}+c_{4} t +c_{1} \]

Problem 5865

ODE	\[ \boxed {x x^{\prime \prime }-{x^{\prime }}^{2}=0} \]
program solution	\[ x = c_{2} {\mathrm e}^{c_{1} t} \] Verified OK.
Maple solution	\begin{align*} x \left (t \right ) &= 0 \\ x \left (t \right ) &= {\mathrm e}^{c_{1} t} c_{2} \\ \end{align*}

Problem 5866

ODE	\[ \boxed {y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime }+3 y^{\prime \prime }-4 y^{\prime }-4 y=f \left (x \right )} \]
program solution	\[ y = \left ({\mathrm e}^{3 x} c_{4} +c_{1} {\mathrm e}^{x}+c_{3} x +c_{2} \right ) {\mathrm e}^{-2 x}+\frac {\left (\left (\int f \left (x \right ) {\mathrm e}^{-x}d x \right ) {\mathrm e}^{3 x}+6 x \left (\int f \left (x \right ) {\mathrm e}^{2 x}d x \right )-9 \left (\int f \left (x \right ) {\mathrm e}^{x}d x \right ) {\mathrm e}^{x}-2 \left (\int \left (-4+3 x \right ) f \left (x \right ) {\mathrm e}^{2 x}d x \right )\right ) {\mathrm e}^{-2 x}}{18} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\left (\int f \left (x \right ) {\mathrm e}^{-x}d x \right ) {\mathrm e}^{3 x}+18 \,{\mathrm e}^{3 x} c_{1} -9 \left (\int f \left (x \right ) {\mathrm e}^{x}d x \right ) {\mathrm e}^{x}+6 x \left (\int f \left (x \right ) {\mathrm e}^{2 x}d x \right )+18 c_{3} {\mathrm e}^{x}+18 c_{4} x -2 \left (\int f \left (x \right ) \left (3 x -4\right ) {\mathrm e}^{2 x}d x \right )+18 c_{2} \right ) {\mathrm e}^{-2 x}}{18} \]

Problem 5867

ODE	\[ \boxed {u^{\prime \prime }-\left (2 x +1\right ) u^{\prime }+\left (x^{2}+x -1\right ) u=0} \]
program solution	\[ u = c_{1} {\mathrm e}^{\frac {x^{2}}{2}}+c_{2} {\mathrm e}^{\frac {x \left (x +2\right )}{2}} \] Verified OK.
Maple solution	\[ u \left (x \right ) = {\mathrm e}^{\frac {x^{2}}{2}} c_{1} +c_{2} {\mathrm e}^{\frac {x \left (x +2\right )}{2}} \]

Problem 5868

ODE	\[ \boxed {y^{\prime \prime }+6 y^{\prime }+9 y=50 \,{\mathrm e}^{2 x}} \]
program solution	\[ y = {\mathrm e}^{-3 x} \left (c_{2} x +c_{1} \right )+2 \,{\mathrm e}^{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (2 \,{\mathrm e}^{5 x}+c_{1} x +c_{2} \right ) {\mathrm e}^{-3 x} \]

Problem 5869

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+4 y=50 \,{\mathrm e}^{2 x}} \]
program solution	\[ y = {\mathrm e}^{2 x} \left (c_{2} x +c_{1} \right )+25 x^{2} {\mathrm e}^{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{2 x} \left (c_{1} x +25 x^{2}+c_{2} \right ) \]

Problem 5870

ODE	\[ \boxed {y^{\prime \prime }+3 y^{\prime }+2 y=\cos \left (2 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{-x}-\frac {\cos \left (2 x \right )}{20}+\frac {3 \sin \left (2 x \right )}{20} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -{\mathrm e}^{-2 x} c_{1} +c_{2} {\mathrm e}^{-x}-\frac {\cos \left (2 x \right )}{20}+\frac {3 \sin \left (2 x \right )}{20} \]

Problem 5871

ODE	\[ \boxed {y^{\prime \prime \prime }+6 y^{\prime \prime }+11 y^{\prime }+6 y=2 \sin \left (3 x \right )} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{-2 x}+{\mathrm e}^{-3 x} c_{3} -\frac {\cos \left (3 x \right )}{195}-\frac {8 \sin \left (3 x \right )}{195} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\cos \left (3 x \right )}{195}-\frac {8 \sin \left (3 x \right )}{195}+c_{1} {\mathrm e}^{-3 x}+c_{2} {\mathrm e}^{-2 x}+c_{3} {\mathrm e}^{-x} \]

Problem 5872

ODE	\[ \boxed {y^{\prime \prime }+4 y=x^{2}} \]
program solution	\[ y = c_{1} \cos \left (2 x \right )+\frac {c_{2} \sin \left (2 x \right )}{2}+\frac {x^{2}}{4}-\frac {1}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sin \left (2 x \right ) c_{2} +\cos \left (2 x \right ) c_{1} +\frac {x^{2}}{4}-\frac {1}{8} \]

Problem 5873

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime }+3 y=x^{3}} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+\frac {c_{2} {\mathrm e}^{3 x}}{2}+\frac {x^{3}}{3}+\frac {4 x^{2}}{3}+\frac {26 x}{9}+\frac {80}{27} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{x} c_{2} +{\mathrm e}^{3 x} c_{1} +\frac {x^{3}}{3}+\frac {4 x^{2}}{3}+\frac {26 x}{9}+\frac {80}{27} \]

Problem 5874

ODE	\[ \boxed {y^{\prime \prime }+2 y^{\prime }+\left (1+\frac {2}{\left (3 x +1\right )^{2}}\right ) y=0} \]
program solution	\[ y = c_{1} \left (\frac {1}{3}+x \right )^{\frac {1}{3}} {\mathrm e}^{-x}+\frac {c_{2} {\mathrm e}^{-x} \left (3 x +1\right )}{\left (\frac {1}{3}+x \right )^{\frac {1}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-x} \left (3 x +1\right )^{\frac {1}{3}} \left (\left (3 x +1\right )^{\frac {1}{3}} c_{2} +c_{1} \right ) \]

Problem 5875

ODE	\[ \boxed {y+\sqrt {x^{2}+y^{2}}-y^{\prime } x=0} \]
program solution	\[ y = -\frac {{\mathrm e}^{-c_{1}} \left ({\mathrm e}^{2 c_{1}}-x^{2}\right )}{2} \] Verified OK.
Maple solution	\[ \frac {-c_{1} x^{2}+y \left (x \right )+\sqrt {x^{2}+y \left (x \right )^{2}}}{x^{2}} = 0 \]

Problem 5876

ODE	\[ \boxed {{y^{\prime }}^{2}+y^{2}=a^{2}} \]
program solution	\[ \arctan \left (\frac {y}{\sqrt {a^{2}-y^{2}}}\right ) = x +c_{1} \] Verified OK. \[ \arctan \left (\frac {\sqrt {a^{2}-y^{2}}}{y}\right ) = x +c_{2} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -a \\ y \left (x \right ) &= a \\ y \left (x \right ) &= -\tan \left (-x +c_{1} \right ) \sqrt {\cos \left (-x +c_{1} \right )^{2} a^{2}} \\ y \left (x \right ) &= \tan \left (-x +c_{1} \right ) \sqrt {\cos \left (-x +c_{1} \right )^{2} a^{2}} \\ \end{align*}

Problem 5877

ODE	\[ \boxed {x^{2} y^{\prime \prime }-2 y^{\prime } x +y \left (x^{2}+2\right )=0} \]
program solution	\[ y = \cos \left (x \right ) c_{1} x +\sin \left (x \right ) c_{2} x \] Verified OK.
Maple solution	\[ y \left (x \right ) = x \left (c_{1} \sin \left (x \right )+\cos \left (x \right ) c_{2} \right ) \]

Problem 5878

ODE	\[ \boxed {y^{\prime \prime }+\frac {2 y^{\prime }}{x}-\frac {2 y}{\left (1+x \right )^{2}}=0} \]
program solution	\[ y = -\frac {c_{1}}{3 x \left (1+x \right )}+\frac {c_{2} \left (1+x \right )^{2}}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (x^{3}+3 x^{2}+3 x \right ) c_{2} +c_{1}}{x \left (x +1\right )} \]

Problem 5879

ODE	\[ \boxed {\left (x^{2} y^{2}+1\right ) y+\left (x^{2} y^{2}-1\right ) x y^{\prime }=0} \]
program solution	\[ y = {\mathrm e}^{-\frac {\operatorname {LambertW}\left (-x^{4} {\mathrm e}^{-2 c_{1}}\right )}{2}-c_{1}} x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{-2 c_{1}} x}{\sqrt {-\frac {x^{4} {\mathrm e}^{-4 c_{1}}}{\operatorname {LambertW}\left (-x^{4} {\mathrm e}^{-4 c_{1}}\right )}}} \]

Problem 5880

ODE	\[ \boxed {2 x^{3} y^{2}-y+\left (2 x^{2} y^{3}-x \right ) y^{\prime }=0} \]
program solution	\[ \frac {y x^{3}+1}{x y}+y^{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {12^{\frac {1}{3}} \left (-{\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {2}{3}}+x^{2} 12^{\frac {1}{3}} \left (x^{2}-c_{1} \right )\right )}{6 {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= -\frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} \left (\left (1+i \sqrt {3}\right ) {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {2}{3}}+2^{\frac {2}{3}} x^{2} \left (x^{2}-c_{1} \right ) \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right )\right )}{12 {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {2^{\frac {2}{3}} 3^{\frac {1}{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {2}{3}}+2^{\frac {2}{3}} x^{2} \left (x^{2}-c_{1} \right ) \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right )\right )}{12 {\left (\left (-9+\sqrt {12 x^{8}-36 c_{1} x^{6}+36 c_{1}^{2} x^{4}-12 c_{1}^{3} x^{2}+81}\right ) x^{2}\right )}^{\frac {1}{3}} x} \\ \end{align*}

Problem 5881

ODE	\[ \boxed {\frac {1}{y}+\sec \left (\frac {y}{x}\right )-\frac {x y^{\prime }}{y^{2}}=0} \]
program solution	\[ \int _{}^{\frac {y}{x}}\frac {\cos \left (\textit {\_a} \right )}{\textit {\_a}^{2}}d \textit {\_a} = x +c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (\textit {\_Z} \,\operatorname {Si}\left (\textit {\_Z} \right )+\textit {\_Z} c_{1} +\textit {\_Z} x +\cos \left (\textit {\_Z} \right )\right ) x \]

Problem 5882

ODE	\[ \boxed {\phi ^{\prime }-\frac {\phi ^{2}}{2}-\phi \cot \left (\theta \right )=0} \]
program solution	\[ \phi = \frac {2 \sin \left (\theta \right )}{c_{3} +\cos \left (\theta \right )} \] Verified OK.
Maple solution	\[ \phi \left (\theta \right ) = \frac {2 \sin \left (\theta \right )}{\cos \left (\theta \right )+2 c_{1}} \]

Problem 5883

ODE	\[ \boxed {u^{\prime \prime }-\cot \left (\theta \right ) u^{\prime }=0} \]
program solution	\[ u = -c_{1} \cos \left (\theta \right )+c_{2} \] Verified OK.
Maple solution	\[ u \left (\theta \right ) = c_{1} +\cos \left (\theta \right ) c_{2} \]

Problem 5884

ODE	\[ \boxed {\left (\phi ^{\prime }-\frac {\phi ^{2}}{2}\right ) \sin \left (\theta \right )^{2}-\cos \left (\theta \right ) \phi \sin \left (\theta \right )=\frac {\cos \left (2 \theta \right )}{2}+1} \]
program solution	\[ \phi = \frac {\left (-\cot \left (\theta \right )+1\right ) \left (\sin \left (\theta \right )+i \cos \left (\theta \right )\right )^{-\frac {i}{2}}-c_{3} \left (\sin \left (\theta \right )+i \cos \left (\theta \right )\right )^{\frac {i}{2}} \left (\cot \left (\theta \right )+1\right )}{c_{3} \left (\sin \left (\theta \right )+i \cos \left (\theta \right )\right )^{\frac {i}{2}}+\left (\sin \left (\theta \right )+i \cos \left (\theta \right )\right )^{-\frac {i}{2}}} \] Verified OK.
Maple solution	\[ \phi \left (\theta \right ) = \frac {-\sinh \left (\frac {\theta }{2}\right ) c_{1} -\cosh \left (\frac {\theta }{2}\right )}{\cosh \left (\frac {\theta }{2}\right ) c_{1} +\sinh \left (\frac {\theta }{2}\right )}-\cot \left (\theta \right ) \]

Problem 5885

ODE	Solve \begin {gather*} \boxed {a y^{\prime \prime } y^{\prime \prime \prime }-\sqrt {1+\left (y^{\prime \prime }\right )^{2}}=0} \end {gather*}
program solution	N/A
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {1}{2} i x^{2}+c_{1} x +c_{2} \\ y \left (x \right ) &= \frac {1}{2} i x^{2}+c_{1} x +c_{2} \\ y \left (x \right ) &= \frac {\left (2 a^{2}+\left (x +c_{1} \right )^{2}\right ) \sqrt {-a^{2}+c_{1}^{2}+2 c_{1} x +x^{2}}-3 \left (a \left (x +c_{1} \right ) \ln \left (c_{1} +x +\sqrt {\left (c_{1} +a +x \right ) \left (c_{1} -a +x \right )}\right )-2 c_{2} x -2 c_{3} \right ) a}{6 a} \\ y \left (x \right ) &= \frac {\left (-2 a^{2}-\left (x +c_{1} \right )^{2}\right ) \sqrt {-a^{2}+c_{1}^{2}+2 c_{1} x +x^{2}}+3 a \left (a \left (x +c_{1} \right ) \ln \left (c_{1} +x +\sqrt {\left (c_{1} +a +x \right ) \left (c_{1} -a +x \right )}\right )+2 c_{2} x +2 c_{3} \right )}{6 a} \\ \end{align*}

Problem 5886

ODE	\[ \boxed {a^{2} y^{\prime \prime \prime \prime }-y^{\prime \prime }=0} \]
program solution	\[ y = c_{2} x +c_{1} +{\mathrm e}^{-\frac {x}{a}} c_{3} +{\mathrm e}^{\frac {x}{a}} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} +c_{2} x +c_{3} {\mathrm e}^{\frac {x}{a}}+c_{4} {\mathrm e}^{-\frac {x}{a}} \]

Problem 5887

ODE	\[ \boxed {y \,{\mathrm e}^{y x}+x \,{\mathrm e}^{y x} y^{\prime }=0} \]
program solution	\[ y = \frac {{\mathrm e}^{-c_{1}}}{x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{x} \]

Problem 5888

ODE	\[ \boxed {-2 y x +{\mathrm e}^{y}+\left (y-x^{2}+{\mathrm e}^{y} x \right ) y^{\prime }=-x} \]
program solution	\[ {\mathrm e}^{y} x -\left (y-\frac {1}{2}\right ) x^{2}+\frac {y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\[ -x^{2} y \left (x \right )+x \,{\mathrm e}^{y \left (x \right )}+\frac {x^{2}}{2}+\frac {y \left (x \right )^{2}}{2}+c_{1} = 0 \]

Problem 5889

ODE	\[ \boxed {y^{\prime \prime }-\frac {y^{\prime }}{\sqrt {x}}+\frac {\left (x +\sqrt {x}-8\right ) y}{4 x^{2}}=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{\sqrt {x}}}{x}+\frac {c_{2} x^{2} {\mathrm e}^{\sqrt {x}}}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {{\mathrm e}^{\sqrt {x}} \left (c_{2} x^{3}+c_{1} \right )}{x} \]

Problem 5890

ODE	\[ \boxed {\left (-x^{2}+1\right ) z^{\prime \prime }+\left (1-3 x \right ) z^{\prime }+k z=0} \]
program solution
Maple solution	\[ z \left (x \right ) = c_{1} \left (x +1\right )^{-1-\sqrt {k +1}} \operatorname {hypergeom}\left (\left [\sqrt {k +1}, 1+\sqrt {k +1}\right ], \left [1+2 \sqrt {k +1}\right ], \frac {2}{x +1}\right )+c_{2} \left (x +1\right )^{-1+\sqrt {k +1}} \operatorname {hypergeom}\left (\left [-\sqrt {k +1}, 1-\sqrt {k +1}\right ], \left [1-2 \sqrt {k +1}\right ], \frac {2}{x +1}\right ) \]

Problem 5891

ODE	\[ \boxed {\left (-x^{2}+1\right ) \eta ^{\prime \prime }-\left (1+x \right ) \eta ^{\prime }+\left (k +1\right ) \eta =0} \]
program solution
Maple solution	\[ \eta \left (x \right ) = c_{1} \left (x +1\right )^{\sqrt {k +1}} \operatorname {hypergeom}\left (\left [-\sqrt {k +1}, 1-\sqrt {k +1}\right ], \left [1-2 \sqrt {k +1}\right ], \frac {2}{x +1}\right )+c_{2} \left (x +1\right )^{-\sqrt {k +1}} \operatorname {hypergeom}\left (\left [\sqrt {k +1}, 1+\sqrt {k +1}\right ], \left [1+2 \sqrt {k +1}\right ], \frac {2}{x +1}\right ) \]

Problem 5892

ODE	\[ \boxed {y^{2}-2 x y y^{\prime }=-x^{2}} \]
program solution	\[ x -\frac {y^{2}}{x} = c_{1} \] Verified OK.
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 5893

ODE	\[ \boxed {-y^{2}+2 x y y^{\prime }=-x^{2}} \]
program solution	\[ x +\frac {y^{2}}{x} = c_{1} \] Verified OK.
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 5894

ODE	\[ \boxed {y^{\prime } x -y-y^{2}=x^{2}} \]
program solution	\[ y = \frac {\left (-c_{3} \cos \left (x \right )+\sin \left (x \right )\right ) x}{c_{3} \sin \left (x \right )+\cos \left (x \right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \tan \left (x +c_{1} \right ) x \]

Problem 5895

ODE	\[ \boxed {y^{\prime } x -y-x \sqrt {-y^{2}+x^{2}}\, y^{\prime }=0} \]
program solution
Maple solution	\[ y \left (x \right )-\arctan \left (\frac {y \left (x \right )}{\sqrt {x^{2}-y \left (x \right )^{2}}}\right )-c_{1} = 0 \]

Problem 5896

ODE	\[ \boxed {y y^{\prime }+y-y^{\prime } x=-x} \]
program solution	\[ \frac {\ln \left (x^{2}+y^{2}\right )}{2}+\arctan \left (\frac {x}{y}\right ) = c_{1} \] Verified OK.
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 5897

ODE	\[ \boxed {y y^{\prime \prime }-{y^{\prime }}^{2}-y^{\prime } y^{2}=0} \]
program solution	\[ y = -\frac {c_{1} c_{2} {\mathrm e}^{c_{1} x}}{-1+c_{2} {\mathrm e}^{c_{1} x}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 0 \\ y \left (x \right ) &= -\frac {c_{1} {\mathrm e}^{\left (x +c_{2} \right ) c_{1}}}{-1+{\mathrm e}^{\left (x +c_{2} \right ) c_{1}}} \\ \end{align*}

Problem 5898

ODE	\begin {align*} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-18 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=2 x_{1} \left (t \right )-9 x_{2} \left (t \right ) \end {align*} With initial conditions \[ [x_{1} \left (0\right ) = 2, x_{2} \left (0\right ) = 1] \]
program solution
Maple solution	\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (-6 t +2\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (-36 t +18\right )}{18} \\ \end{align*}

Problem 5899

ODE	\begin {align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )+3 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=5 x_{1} \left (t \right )+3 x_{2} \left (t \right ) \end {align*}
program solution
Maple solution	\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{6 t} c_{1} +c_{2} {\mathrm e}^{-2 t} \\ x_{2} \left (t \right ) &= \frac {5 \,{\mathrm e}^{6 t} c_{1}}{3}-c_{2} {\mathrm e}^{-2 t} \\ \end{align*}

Problem 5900

ODE	\begin {align*} x_{1}^{\prime }\left (t \right )&=-x_{1} \left (t \right )+3 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-3 x_{1} \left (t \right )+5 x_{2} \left (t \right ) \end {align*} With initial conditions \[ [x_{1} \left (0\right ) = 1, x_{2} \left (0\right ) = 2] \]
program solution
Maple solution	\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (3 t +1\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (9 t +6\right )}{3} \\ \end{align*}