2.17.10 Problems 901 to 1000

Problem 901

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

Problem 902

ODE	\[ \boxed {y^{\prime }+\frac {y}{x}=\frac {7}{x^{2}}+3} \]
program solution	\[ y = \frac {3 x^{2}+14 \ln \left (x \right )+2 c_{1}}{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\frac {3 x^{2}}{2}+7 \ln \left (x \right )+c_{1}}{x} \]

Problem 903

ODE	\[ \boxed {y^{\prime }+\frac {4 y}{x -1}=\frac {1}{\left (x -1\right )^{5}}+\frac {\sin \left (x \right )}{\left (x -1\right )^{4}}} \]
program solution	\[ y = -\frac {\cos \left (x \right )-\ln \left (x -1\right )-c_{1}}{x^{4}-4 x^{3}+6 x^{2}-4 x +1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-\cos \left (x \right )+\ln \left (x -1\right )+c_{1}}{\left (x -1\right )^{4}} \]

Problem 904

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

Problem 905

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

Problem 906

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

Problem 907

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

Problem 908

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

Problem 909

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

Problem 910

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

Problem 911

ODE	\[ \boxed {y^{\prime }+7 y={\mathrm e}^{3 x}} \] With initial conditions \begin {align*} [y \left (0\right ) = 0] \end {align*}
program solution	\[ y = \frac {{\mathrm e}^{3 x}}{10}-\frac {{\mathrm e}^{-7 x}}{10} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left ({\mathrm e}^{10 x}-1\right ) {\mathrm e}^{-7 x}}{10} \]

Problem 912

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

Problem 913

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

Problem 914

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

Problem 915

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

Problem 916

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

Problem 917

ODE	\[ \boxed {2 y+y^{\prime } x=8 x^{2}} \] With initial conditions \begin {align*} [y \left (1\right ) = 3] \end {align*}
program solution	\[ y = \frac {2 x^{4}+1}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2 x^{4}+1}{x^{2}} \]

Problem 918

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

Problem 919

ODE	\[ \boxed {2 y x +y^{\prime }=x} \] With initial conditions \begin {align*} [y \left (0\right ) = 3] \end {align*}
program solution	\[ y = \frac {1}{2}+\frac {5 \,{\mathrm e}^{-x^{2}}}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {1}{2}+\frac {5 \,{\mathrm e}^{-x^{2}}}{2} \]

Problem 920

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

Problem 921

ODE	\[ \boxed {\left (2+x \right ) y^{\prime }+4 y=\frac {2 x^{2}+1}{x \left (2+x \right )^{3}}} \] With initial conditions \begin {align*} [y \left (-1\right ) = 2] \end {align*}
program solution	\[ y = \frac {x^{2}+\ln \left (x \right )-i \pi +1}{x^{4}+8 x^{3}+24 x^{2}+32 x +16} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{2}+\ln \left (x \right )+1-i \pi }{\left (2+x \right )^{4}} \]

Problem 922

ODE	\[ \boxed {\left (x^{2}-1\right ) y^{\prime }-2 y x=x \left (x^{2}-1\right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 4] \end {align*}
program solution	\[ y = -\frac {i \pi \,x^{2}}{2}+\frac {\ln \left (x -1\right ) x^{2}}{2}+\frac {\ln \left (x +1\right ) x^{2}}{2}+\frac {i \pi }{2}-4 x^{2}-\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2}+4 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (i \pi -\ln \left (x +1\right )-\ln \left (x -1\right )+8\right ) \left (x^{2}-1\right )}{2} \]

Problem 923

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

Problem 924

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

Problem 925

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

Problem 926

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

Problem 927

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

Problem 928

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

Problem 929

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

Problem 930

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

Problem 931

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

Problem 932

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

Problem 933

ODE	\[ \boxed {y^{\prime }-\left (1+y^{2}\right ) x^{2}=0} \]
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 934

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

Problem 935

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

Problem 936

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

Problem 937

ODE	\[ \boxed {y^{\prime }-\frac {x^{2}+3 x +2}{y-2}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 4] \end {align*}
program solution	\[ -\frac {x^{3}}{3}-\frac {3 x^{2}}{2}+\frac {y^{2}}{2}-2 x -2 y = -{\frac {23}{6}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 2+\frac {\sqrt {6 x^{3}+27 x^{2}+36 x -33}}{3} \]

Problem 938

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

Problem 939

ODE	\[ \boxed {\left (3 y^{2}+4 y\right ) y^{\prime }=-2 x -\cos \left (x \right )} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ -x^{2}-\sin \left (x \right )-y^{3}-2 y^{2} = -3 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (260-108 x^{2}-108 \sin \left (x \right )+12 \sqrt {441-390 x^{2}-390 \sin \left (x \right )+81 x^{4}+162 \sin \left (x \right ) x^{2}+81 \sin \left (x \right )^{2}}\right )^{\frac {1}{3}}}{6}+\frac {8}{3 \left (260-108 x^{2}-108 \sin \left (x \right )+12 \sqrt {441-390 x^{2}-390 \sin \left (x \right )+81 x^{4}+162 \sin \left (x \right ) x^{2}+81 \sin \left (x \right )^{2}}\right )^{\frac {1}{3}}}-\frac {2}{3} \]

Problem 940

ODE	\[ \boxed {y^{\prime }+\frac {\left (1+y\right ) \left (y-1\right ) \left (y-2\right )}{x +1}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 0] \end {align*}
program solution	\[ y = \frac {{\operatorname {RootOf}\left (-27+\left (c_{4}^{6} x^{6}+6 c_{4}^{6} x^{5}+15 c_{4}^{6} x^{4}+20 c_{4}^{6} x^{3}+15 c_{4}^{6} x^{2}+6 c_{4}^{6} x +c_{4}^{6}-1\right ) \textit {\_Z}^{9}+\left (9 c_{4}^{6} x^{6}+54 c_{4}^{6} x^{5}+135 c_{4}^{6} x^{4}+180 c_{4}^{6} x^{3}+135 c_{4}^{6} x^{2}+54 c_{4}^{6} x +9 c_{4}^{6}-9\right ) \textit {\_Z}^{6}-27 \textit {\_Z}^{3}\right )}^{3}}{3}+\frac {10}{3} \] Warning, solution could not be verified
Maple solution	\begin{align*} y \left (x \right ) &= \frac {8 \operatorname {RootOf}\left (512 \textit {\_Z}^{6}+\left (96 \,2^{\frac {1}{3}} x^{2}+192 x 2^{\frac {1}{3}}+96 \,2^{\frac {1}{3}}\right ) \textit {\_Z}^{4}-x^{6}-6 x^{5}-15 x^{4}-20 x^{3}-15 x^{2}-6 x -257\right )^{2}-2^{\frac {1}{3}} \left (x +1\right )^{2}}{8 \operatorname {RootOf}\left (512 \textit {\_Z}^{6}+\left (96 \,2^{\frac {1}{3}} x^{2}+192 x 2^{\frac {1}{3}}+96 \,2^{\frac {1}{3}}\right ) \textit {\_Z}^{4}-x^{6}-6 x^{5}-15 x^{4}-20 x^{3}-15 x^{2}-6 x -257\right )^{2}} \\ y \left (x \right ) &= \frac {16 \operatorname {RootOf}\left (512 \textit {\_Z}^{6}+\left (48 i x^{2} \sqrt {3}\, 2^{\frac {1}{3}}+96 i x \sqrt {3}\, 2^{\frac {1}{3}}+48 i \sqrt {3}\, 2^{\frac {1}{3}}-48 \,2^{\frac {1}{3}} x^{2}-96 x 2^{\frac {1}{3}}-48 \,2^{\frac {1}{3}}\right ) \textit {\_Z}^{4}-x^{6}-6 x^{5}-15 x^{4}-20 x^{3}-15 x^{2}-6 x -257\right )^{2}-\left (x +1\right )^{2} \left (i \sqrt {3}-1\right ) 2^{\frac {1}{3}}}{16 \operatorname {RootOf}\left (512 \textit {\_Z}^{6}+\left (48 i x^{2} \sqrt {3}\, 2^{\frac {1}{3}}+96 i x \sqrt {3}\, 2^{\frac {1}{3}}+48 i \sqrt {3}\, 2^{\frac {1}{3}}-48 \,2^{\frac {1}{3}} x^{2}-96 x 2^{\frac {1}{3}}-48 \,2^{\frac {1}{3}}\right ) \textit {\_Z}^{4}-x^{6}-6 x^{5}-15 x^{4}-20 x^{3}-15 x^{2}-6 x -257\right )^{2}} \\ y \left (x \right ) &= \frac {16 \operatorname {RootOf}\left (512 \textit {\_Z}^{6}+\left (-48 i x^{2} \sqrt {3}\, 2^{\frac {1}{3}}-96 i x \sqrt {3}\, 2^{\frac {1}{3}}-48 i \sqrt {3}\, 2^{\frac {1}{3}}-48 \,2^{\frac {1}{3}} x^{2}-96 x 2^{\frac {1}{3}}-48 \,2^{\frac {1}{3}}\right ) \textit {\_Z}^{4}-x^{6}-6 x^{5}-15 x^{4}-20 x^{3}-15 x^{2}-6 x -257\right )^{2}+\left (1+i \sqrt {3}\right ) \left (x +1\right )^{2} 2^{\frac {1}{3}}}{16 \operatorname {RootOf}\left (512 \textit {\_Z}^{6}+\left (-48 i x^{2} \sqrt {3}\, 2^{\frac {1}{3}}-96 i x \sqrt {3}\, 2^{\frac {1}{3}}-48 i \sqrt {3}\, 2^{\frac {1}{3}}-48 \,2^{\frac {1}{3}} x^{2}-96 x 2^{\frac {1}{3}}-48 \,2^{\frac {1}{3}}\right ) \textit {\_Z}^{4}-x^{6}-6 x^{5}-15 x^{4}-20 x^{3}-15 x^{2}-6 x -257\right )^{2}} \\ \end{align*}

Problem 941

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

Problem 942

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

Problem 943

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

Problem 944

ODE	\[ \boxed {y^{\prime }+2 x \left (y^{3}-3 y+2\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 3] \end {align*}
program solution	\[ -\frac {x^{2}}{2}+\frac {1}{6 y-6}+\frac {\ln \left (y-1\right )}{18}-\frac {\ln \left (2+y\right )}{18} = \frac {1}{12}+\frac {\ln \left (2\right )}{18}-\frac {\ln \left (5\right )}{18} \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\operatorname {RootOf}\left (18 x^{2} {\mathrm e}^{\textit {\_Z}}-2 \ln \left ({\mathrm e}^{\textit {\_Z}}-3\right ) {\mathrm e}^{\textit {\_Z}}+2 \,{\mathrm e}^{\textit {\_Z}} \ln \left (2\right )-2 \,{\mathrm e}^{\textit {\_Z}} \ln \left (5\right )+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-54 x^{2}+3 \,{\mathrm e}^{\textit {\_Z}}+6 \ln \left ({\mathrm e}^{\textit {\_Z}}-3\right )-6 \ln \left (2\right )+6 \ln \left (5\right )-6 \textit {\_Z} -15\right )}-2 \]

Problem 945

ODE	\[ \boxed {y^{\prime }-\frac {2 x}{1+2 y}=0} \] With initial conditions \begin {align*} [y \left (2\right ) = 0] \end {align*}
program solution	\[ -\frac {x^{2}}{2}+\frac {y}{2}+\frac {y^{2}}{2} = -2 \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {1}{2}+\frac {\sqrt {4 x^{2}-15}}{2} \]

Problem 946

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

Problem 947

ODE	\[ \boxed {y y^{\prime }=-x} \] With initial conditions \begin {align*} [y \left (3\right ) = -4] \end {align*}
program solution	\[ -\frac {x^{2}}{2}-\frac {y^{2}}{2} = -{\frac {25}{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\sqrt {-x^{2}+25} \]

Problem 948

ODE	\[ \boxed {y^{\prime }+x^{2} \left (1+y\right ) \left (y-2\right )^{2}=0} \]
program solution	\[ -\frac {\ln \left (3+y\right )}{9}+\frac {1}{3 y}+\frac {\ln \left (y\right )}{9}-\frac {x^{3}}{3}-c_{2} = 0 \] Verified OK.
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\operatorname {RootOf}\left (3 x^{3} {\mathrm e}^{\textit {\_Z}}+\ln \left ({\mathrm e}^{\textit {\_Z}}+3\right ) {\mathrm e}^{\textit {\_Z}}+9 c_{1} {\mathrm e}^{\textit {\_Z}}-\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-3\right )}+2 \]

Problem 949

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

Problem 950

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

Problem 951

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

Problem 952

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

Problem 953

ODE	\[ \boxed {y^{\prime }-a y+b y^{2}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = \operatorname {y0}] \end {align*}
program solution	\[ \frac {\ln \left (y\right )-\ln \left (b y-a \right )}{a} = x +\frac {\ln \left (\operatorname {y0} \right )-\ln \left (b \operatorname {y0} -a \right )}{a} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {a \operatorname {y0}}{\left (-\operatorname {y0} b +a \right ) {\mathrm e}^{-a x}+\operatorname {y0} b} \]

Problem 954

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

Problem 955

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

Problem 956

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

Problem 957

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

Problem 958

ODE	\[ \boxed {y^{\prime }-\frac {x^{2}+y^{2}}{\sin \left (x \right )}=0} \]
program solution	\[ y = -\frac {\left (\frac {d}{d x}\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )+\cot \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )+\csc \left (x \right )^{2} x^{2} \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )\right ) \sin \left (x \right )}{\operatorname {DESol}\left (\left \{\textit {\_Y}^{\prime \prime }\left (x \right )+\cot \left (x \right ) \textit {\_Y}^{\prime }\left (x \right )+\csc \left (x \right )^{2} x^{2} \textit {\_Y} \left (x \right )\right \}, \left \{\textit {\_Y} \left (x \right )\right \}\right )} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 959

ODE	\[ \boxed {y^{\prime }-\frac {y+{\mathrm e}^{x}}{x^{2}+y^{2}}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 960

ODE	\[ \boxed {y^{\prime }-\tan \left (y x \right )=0} \]
program solution
Maple solution	\[ y \left (x \right ) = -i \operatorname {RootOf}\left (-\operatorname {erf}\left (\frac {\left (-x +\textit {\_Z} \right ) \sqrt {2}}{2}\right ) \sqrt {\pi }-\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {2}\, \left (x +\textit {\_Z} \right )}{2}\right )+\sqrt {2}\, c_{1} \right ) \]

Problem 961

ODE	\[ \boxed {y^{\prime }-\frac {x^{2}+y^{2}}{\ln \left (y x \right )}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 962

ODE	\[ \boxed {y^{\prime }-\left (x^{2}+y^{2}\right ) y^{\frac {1}{3}}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 963

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

Problem 964

ODE	\[ \boxed {y^{\prime }-\ln \left (1+x^{2}+y^{2}\right )=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 965

ODE	\[ \boxed {y^{\prime }-\frac {2 x +3 y}{x -4 y}=0} \]
program solution	\[ \frac {\ln \left (2 y^{2}+y x +x^{2}\right )}{2}-\frac {2 \sqrt {7}\, \arctan \left (\frac {\left (4 y+x \right ) \sqrt {7}}{7 x}\right )}{7} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x \left (\sqrt {7}\, \tan \left (\operatorname {RootOf}\left (\sqrt {7}\, \ln \left (\sec \left (\textit {\_Z} \right )^{2} x^{2}\right )+\sqrt {7}\, \ln \left (7\right )-3 \sqrt {7}\, \ln \left (2\right )+2 \sqrt {7}\, c_{1} -4 \textit {\_Z} \right )\right )-1\right )}{4} \]

Problem 966

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

Problem 967

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

Problem 968

ODE	\[ \boxed {y^{\prime }-\left (x^{2}+y^{2}\right )^{2}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 969

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

Problem 970

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

Problem 971

ODE	\[ \boxed {y^{\prime }-y^{\frac {2}{5}}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ \frac {5 y^{\frac {3}{5}}}{3} = x +\frac {5}{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (3 x +5\right ) \left (\frac {3 x}{5}+1\right )^{\frac {2}{3}}}{5} \]

Problem 972

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

Problem 973

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

Problem 974

ODE	\[ \boxed {y^{\prime }-3 x \left (y-1\right )^{\frac {1}{3}}=0} \] With initial conditions \begin {align*} [y \left (3\right ) = -7] \end {align*}
program solution	\[ y = 1+\left (2 i \sqrt {3}+x^{2}-11\right )^{\frac {3}{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = 1+\left (-11+2 i \sqrt {3}+x^{2}\right )^{\frac {3}{2}} \]

Problem 975

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

Problem 976

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

Problem 977

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

Problem 978

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

Problem 979

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

Problem 980

ODE	\[ \boxed {7 y^{\prime } x -2 y+\frac {x^{2}}{y^{6}}=0} \]
program solution	\[ y = \left (-x^{2} \left (\ln \left (x \right )-c_{1} \right )\right )^{\frac {1}{7}} \] Verified OK. \[ y = \left (-1\right )^{\frac {2}{7}} \left (x^{2} \left (-\ln \left (x \right )+c_{1} \right )\right )^{\frac {1}{7}} \] Verified OK. \[ y = \left (-1\right )^{\frac {4}{7}} \left (x^{2} \left (-\ln \left (x \right )+c_{1} \right )\right )^{\frac {1}{7}} \] Verified OK. \[ y = \left (-1\right )^{\frac {6}{7}} \left (x^{2} \left (-\ln \left (x \right )+c_{1} \right )\right )^{\frac {1}{7}} \] Verified OK. \[ y = -\left (-1\right )^{\frac {1}{7}} \left (x^{2} \left (-\ln \left (x \right )+c_{1} \right )\right )^{\frac {1}{7}} \] Verified OK. \[ y = -\left (-1\right )^{\frac {3}{7}} \left (x^{2} \left (-\ln \left (x \right )+c_{1} \right )\right )^{\frac {1}{7}} \] Verified OK. \[ y = -\left (-1\right )^{\frac {5}{7}} \left (x^{2} \left (-\ln \left (x \right )+c_{1} \right )\right )^{\frac {1}{7}} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \left (-x^{2} \left (\ln \left (x \right )-c_{1} \right )\right )^{\frac {1}{7}} \\ y \left (x \right ) &= -\left (\left (c_{1} -\ln \left (x \right )\right ) x^{2}\right )^{\frac {1}{7}} \left (-1\right )^{\frac {1}{7}} \\ y \left (x \right ) &= \left (\left (c_{1} -\ln \left (x \right )\right ) x^{2}\right )^{\frac {1}{7}} \left (-1\right )^{\frac {6}{7}} \\ y \left (x \right ) &= -\left (\left (c_{1} -\ln \left (x \right )\right ) x^{2}\right )^{\frac {1}{7}} \left (-1\right )^{\frac {5}{7}} \\ y \left (x \right ) &= \left (\left (c_{1} -\ln \left (x \right )\right ) x^{2}\right )^{\frac {1}{7}} \left (-1\right )^{\frac {2}{7}} \\ y \left (x \right ) &= -\left (\left (c_{1} -\ln \left (x \right )\right ) x^{2}\right )^{\frac {1}{7}} \left (-1\right )^{\frac {3}{7}} \\ y \left (x \right ) &= \left (\left (c_{1} -\ln \left (x \right )\right ) x^{2}\right )^{\frac {1}{7}} \left (-1\right )^{\frac {4}{7}} \\ \end{align*}

Problem 981

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

Problem 982

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

Problem 983

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

Problem 984

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

Problem 985

ODE	\[ \boxed {y^{\prime }-2 y-x y^{3}=0} \] With initial conditions \begin {align*} \left [y \left (0\right ) = 2 \sqrt {2}\right ] \end {align*}
program solution	\[ \frac {1}{y^{2}} = -\frac {x}{2}+\frac {1}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {4}{\sqrt {-8 x +2}} \]

Problem 986

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

Problem 987

ODE	\[ \boxed {y+y^{\prime } x -x^{4} y^{4}=0} \] With initial conditions \begin {align*} \left [y \left (1\right ) = {\frac {1}{2}}\right ] \end {align*}
program solution	\[ \frac {1}{y^{3}} = -3 x^{4}+11 x^{3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-\left (3 x -11\right )^{2}\right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{6 x^{2}-22 x} \]

Problem 988

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

Problem 989

ODE	\[ \boxed {y^{\prime }-4 y-\frac {48 x}{y^{2}}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ y^{3} = -1+2 \,{\mathrm e}^{12 x}-12 x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (2 \,{\mathrm e}^{12 x}-12 x -1\right )^{\frac {1}{3}} \]

Problem 990

ODE	\[ \boxed {y^{\prime } x^{2}+2 y x -y^{3}=0} \] With initial conditions \begin {align*} \left [y \left (1\right ) = \frac {\sqrt {2}}{2}\right ] \end {align*}
program solution	\[ \frac {1}{y^{2}} = \frac {8 x^{5}+2}{5 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sqrt {10}\, \sqrt {4 x^{6}+x}}{8 x^{5}+2} \]

Problem 991

ODE	\[ \boxed {y^{\prime }-y-\sqrt {y}\, x=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 4] \end {align*}
program solution	\[ \sqrt {y} = -2+4 \,{\mathrm e}^{\frac {x}{2}}-x \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-8 x -16\right ) {\mathrm e}^{\frac {x}{2}}+x^{2}+4 x +16 \,{\mathrm e}^{x}+4 \]

Problem 992

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

Problem 993

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

Problem 994

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

Problem 995

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

Problem 996

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

Problem 997

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

Problem 998

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

Problem 999

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

Problem 1000

ODE	\[ \boxed {y^{\prime }-\frac {x^{3}+y^{3}}{x y^{2}}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 3] \end {align*}
program solution	\[ \frac {y^{3}}{3 x^{3}}-\ln \left (x \right ) = 9 \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (3 \ln \left (x \right )+27\right )^{\frac {1}{3}} x \]