2.17.50 Problems 4901 to 5000

Problem 4901

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

Problem 4902

ODE	\[ \boxed {y^{\prime \prime }-2 y^{\prime }+y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{3} x^{3}-\frac {1}{8} x^{4}-\frac {1}{30} x^{5}-\frac {1}{144} x^{6}\right ) y \left (0\right )+\left (x +x^{2}+\frac {1}{2} x^{3}+\frac {1}{6} x^{4}+\frac {1}{24} x^{5}+\frac {1}{120} x^{6}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1-\frac {1}{2} x^{2}-\frac {1}{3} x^{3}-\frac {1}{8} x^{4}-\frac {1}{30} x^{5}\right ) c_{1} +\left (x +x^{2}+\frac {1}{2} x^{3}+\frac {1}{6} x^{4}+\frac {1}{24} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1-\frac {1}{2} x^{2}-\frac {1}{3} x^{3}-\frac {1}{8} x^{4}-\frac {1}{30} x^{5}\right ) y \left (0\right )+\left (x +x^{2}+\frac {1}{2} x^{3}+\frac {1}{6} x^{4}+\frac {1}{24} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 4903

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

Problem 4904

ODE	\[ \boxed {x^{2} y^{\prime \prime }-3 y^{\prime } x +3 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{3} \left (1+O\left (x^{6}\right )\right )+c_{2} x \left (1+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{3} \left (1+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (-2+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 4905

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

Problem 4906

ODE	\[ \boxed {\left (x^{2}+2 x \right ) y^{\prime \prime }-2 \left (x +1\right ) y^{\prime }+2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = c_{1} x^{2} \left (1+O\left (x^{6}\right )\right )+c_{2} \left (1+x +\frac {x^{2}}{4}+O\left (x^{6}\right )\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+\operatorname {O}\left (x^{6}\right )\right ) c_{1} x^{2}+c_{2} \left (-2-2 x -\frac {1}{2} x^{2}+\operatorname {O}\left (x^{6}\right )\right ) \]

Problem 4907

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

Problem 4908

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +2 y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (-x^{2}+1\right ) y \left (0\right )+y^{\prime }\left (0\right ) x +O\left (x^{6}\right ) \] Verified OK. \[ y = \left (-x^{2}+1\right ) c_{1} +c_{2} x +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = y \left (0\right )+D\left (y \right )\left (0\right ) x -y \left (0\right ) x^{2} \]

Problem 4909

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

Problem 4910

ODE	\[ \boxed {y^{\prime \prime }-4 y^{\prime } x +\left (4 x^{2}-2\right ) y=0} \] With the expansion point for the power series method at \(x = 0\).
program solution	\[ y = \left (1+x^{2}+\frac {1}{2} x^{4}+\frac {1}{6} x^{6}\right ) y \left (0\right )+\left (x +x^{3}+\frac {1}{2} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \] Verified OK. \[ y = \left (1+x^{2}+\frac {1}{2} x^{4}\right ) c_{1} +\left (x +x^{3}+\frac {1}{2} x^{5}\right ) c_{2} +O\left (x^{6}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (1+x^{2}+\frac {1}{2} x^{4}\right ) y \left (0\right )+\left (x +x^{3}+\frac {1}{2} x^{5}\right ) D\left (y \right )\left (0\right )+O\left (x^{6}\right ) \]

Problem 4911

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

Problem 4912

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

Problem 4913

ODE	\[ \boxed {y^{\prime }-4 y^{2}+3 y=1} \]
program solution	\[ y = \frac {\left (3 \sqrt {7}+7 \tan \left (\frac {\left (x +c_{1} \right ) \sqrt {7}}{2}\right )\right ) \sqrt {7}}{56} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3}{8}+\frac {\sqrt {7}\, \tan \left (\frac {\left (x +c_{1} \right ) \sqrt {7}}{2}\right )}{8} \]

Problem 4914

ODE	\[ \boxed {s^{\prime }-t \ln \left (s^{2 t}\right )=8 t^{2}} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 4915

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

Problem 4916

ODE	\[ \boxed {\left (y^{2} x +3 y^{2}\right ) y^{\prime }=2 x} \]
program solution	\[ -x +3 \ln \left (x +3\right )+\frac {y^{3}}{6} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \left (-18 \ln \left (x +3\right )+c_{1} +6 x \right )^{\frac {1}{3}} \\ y \left (x \right ) &= -\frac {\left (-18 \ln \left (x +3\right )+c_{1} +6 x \right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{2} \\ y \left (x \right ) &= \frac {\left (-18 \ln \left (x +3\right )+c_{1} +6 x \right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{2} \\ \end{align*}

Problem 4917

ODE	\[ \boxed {s^{2}+s^{\prime }-\frac {s+1}{t s}=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 4918

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

Problem 4919

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

Problem 4920

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

Problem 4921

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

Problem 4922

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

Problem 4923

ODE	\[ \boxed {y^{\prime }-\frac {\sec \left (y\right )^{2}}{x^{2}+1}=0} \]
program solution	\[ -\arctan \left (x \right )+\frac {\sin \left (y\right ) \cos \left (y\right )}{2}+\frac {y}{2} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\arcsin \left (\operatorname {RootOf}\left (\textit {\_Z} +2 x^{2} \textit {\_Z} +\textit {\_Z} \,x^{4}-x^{4} \sin \left (-\textit {\_Z} +4 c_{1} \right )+4 x^{3} \cos \left (-\textit {\_Z} +4 c_{1} \right )+6 x^{2} \sin \left (-\textit {\_Z} +4 c_{1} \right )-4 x \cos \left (-\textit {\_Z} +4 c_{1} \right )-\sin \left (-\textit {\_Z} +4 c_{1} \right )\right )\right )}{2} \]

Problem 4924

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

Problem 4925

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

Problem 4926

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

Problem 4927

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

Problem 4928

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

Problem 4929

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

Problem 4930

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

Problem 4931

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

Problem 4932

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

Problem 4933

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

Problem 4934

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

Problem 4935

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

Problem 4936

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

Problem 4937

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

Problem 4938

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

Problem 4939

ODE	\[ \boxed {y^{\prime }-\frac {{\mathrm e}^{x^{2}}}{y^{2}}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ -\frac {\sqrt {\pi }\, \operatorname {erfi}\left (x \right )}{2}+\frac {y^{3}}{3} = {\frac {1}{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (8+12 \sqrt {\pi }\, \operatorname {erfi}\left (x \right )\right )^{\frac {1}{3}}}{2} \]

Problem 4940

ODE	\[ \boxed {y^{\prime }-\sqrt {\sin \left (x \right )+1}\, \left (1+y^{2}\right )=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ y = \frac {\cos \left (-2+2 \sqrt {\sin \left (x \right )+1}\, \sec \left (x \right )-2 \sqrt {\sin \left (x \right )+1}\, \tan \left (x \right )+\frac {\pi }{4}\right )}{\sin \left (-2+2 \sqrt {\sin \left (x \right )+1}\, \sec \left (x \right )-2 \sqrt {\sin \left (x \right )+1}\, \tan \left (x \right )+\frac {\pi }{4}\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \tan \left (\sqrt {2}\, \left (\int _{0}^{x}\operatorname {csgn}\left (\sin \left (\frac {\pi }{4}+\frac {\textit {\_z1}}{2}\right )\right ) \sin \left (\frac {\pi }{4}+\frac {\textit {\_z1}}{2}\right )d \textit {\_z1} \right )+\frac {\pi }{4}\right ) \]

Problem 4941

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

Problem 4942

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

Problem 4943

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

Problem 4944

ODE	\[ \boxed {y^{\prime }-\left (x -3\right ) \left (y+1\right )^{\frac {2}{3}}=0} \]
program solution	\[ y = \frac {1}{216} x^{6}+\frac {1}{36} x^{4} c_{1} -\frac {1}{12} x^{5}+\frac {1}{18} c_{1}^{2} x^{2}-\frac {1}{3} c_{1} x^{3}+\frac {1}{2} x^{4}+\frac {1}{27} c_{1}^{3}-\frac {1}{3} c_{1}^{2} x +c_{1} x^{2}-x^{3}-1 \] Verified OK.
Maple solution	\[ \frac {x^{2}}{2}-3 x -3 \left (y \left (x \right )+1\right )^{\frac {1}{3}}+c_{1} = 0 \]

Problem 4945

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

Problem 4946

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

Problem 4947

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

Problem 4948

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

Problem 4949

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

Problem 4950

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

Problem 4951

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

Problem 4952

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

Problem 4953

ODE	\[ \boxed {-{\mathrm e}^{t} y^{\prime }-y \ln \left (t \right )=-3 t} \]
program solution	\[ \int _{}^{t}\left (-y \ln \left (\textit {\_a} \right )+3 \textit {\_a} \right ) {\mathrm e}^{-\textit {\_a} -\operatorname {expIntegral}_{1}\left (\textit {\_a} \right )} \textit {\_a}^{-{\mathrm e}^{-\textit {\_a}}}d \textit {\_a} +\left (-t^{-{\mathrm e}^{-t}} {\mathrm e}^{-\operatorname {expIntegral}_{1}\left (t \right )}+\int _{}^{t}\ln \left (\textit {\_a} \right ) {\mathrm e}^{-\textit {\_a} -\operatorname {expIntegral}_{1}\left (\textit {\_a} \right )} \textit {\_a}^{-{\mathrm e}^{-\textit {\_a}}}d \textit {\_a} \right ) y = c_{1} \] Verified OK.
Maple solution	\[ y \left (t \right ) = \left (3 \left (\int t^{1-{\mathrm e}^{-t}} {\mathrm e}^{-t -\operatorname {expIntegral}_{1}\left (t \right )}d t \right )+c_{1} \right ) t^{{\mathrm e}^{-t}} {\mathrm e}^{\operatorname {expIntegral}_{1}\left (t \right )} \]

Problem 4954

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

Problem 4955

ODE	\[ \boxed {3 r-r^{\prime }=-\theta ^{3}} \]
program solution	\[ r = -\frac {\left (9 \theta ^{3} {\mathrm e}^{-3 \theta }+9 \theta ^{2} {\mathrm e}^{-3 \theta }+6 \,{\mathrm e}^{-3 \theta } \theta +2 \,{\mathrm e}^{-3 \theta }+27 c_{1} \right ) {\mathrm e}^{3 \theta }}{27} \] Verified OK.
Maple solution	\[ r \left (\theta \right ) = -\frac {\theta ^{2}}{3}-\frac {\theta ^{3}}{3}-\frac {2 \theta }{9}-\frac {2}{27}+{\mathrm e}^{3 \theta } c_{1} \]

Problem 4956

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

Problem 4957

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

Problem 4958

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

Problem 4959

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

Problem 4960

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

Problem 4961

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

Problem 4962

ODE	\[ \boxed {y y^{\prime }-5 y^{3}=-2 x} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 4963

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

Problem 4964

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

Problem 4965

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

Problem 4966

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

Problem 4967

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

Problem 4968

ODE	\[ \boxed {t^{2} x^{\prime }+3 x t=t^{4} \ln \left (t \right )+1} \] With initial conditions \begin {align*} [x \left (1\right ) = 0] \end {align*}
program solution	\[ x = \frac {6 t^{6} \ln \left (t \right )-t^{6}+18 t^{2}-17}{36 t^{3}} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {6 t^{6} \ln \left (t \right )-t^{6}+18 t^{2}-17}{36 t^{3}} \]

Problem 4969

ODE	\[ \boxed {y^{\prime }+\frac {3 y}{x}=3 x -2} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
program solution	\[ y = \frac {6 x^{5}-5 x^{4}+9}{10 x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3 x^{2}}{5}-\frac {x}{2}+\frac {9}{10 x^{3}} \]

Problem 4970

ODE	\[ \boxed {\cos \left (x \right ) y^{\prime }+\sin \left (x \right ) y=2 \cos \left (x \right )^{2} x} \] With initial conditions \begin {align*} \left [y \left (\frac {\pi }{4}\right ) = -\frac {15 \sqrt {2}\, \pi ^{2}}{32}\right ] \end {align*}
program solution	\[ y = -\cos \left (x \right ) \pi ^{2}+x^{2} \cos \left (x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (-\pi ^{2}+x^{2}\right ) \cos \left (x \right ) \]

Problem 4971

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

Problem 4972

ODE	\[ \boxed {y^{\prime }+y \sqrt {1+\sin \left (x \right )^{2}}=x} \] With initial conditions \begin {align*} [y \left (0\right ) = 2] \end {align*}
program solution	\[ y = {\mathrm e}^{\sqrt {2}\, \operatorname {EllipticE}\left (\cos \left (x \right ), \frac {\sqrt {2}}{2}\right )} \left (\int _{0}^{x}{\mathrm e}^{-\sqrt {2}\, \operatorname {EllipticE}\left (\cos \left (\textit {\_a} \right ), \frac {\sqrt {2}}{2}\right )} \textit {\_a} d \textit {\_a} \right )+2 \,{\mathrm e}^{-\sqrt {2}\, \left (-\operatorname {EllipticE}\left (\cos \left (x \right ), \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticE}\left (\frac {\sqrt {2}}{2}\right )\right )} \] Verified OK. {positive, sin(_a)::positive, sin(x)::positive}
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\operatorname {csgn}\left (\cos \left (x \right )\right ) \operatorname {EllipticE}\left (\sin \left (x \right ), i\right )} \left (\int _{0}^{x}\textit {\_z1} \,{\mathrm e}^{\operatorname {csgn}\left (\cos \left (\textit {\_z1} \right )\right ) \operatorname {EllipticE}\left (\sin \left (\textit {\_z1} \right ), i\right )}d \textit {\_z1} +2\right ) \]

Problem 4973

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

Problem 4974

ODE	\[ \boxed {y^{\prime }+2 y-\frac {x}{y^{2}}=0} \]
program solution	\[ -\frac {\left (-2 y^{3}+x -\frac {1}{6}\right ) {\mathrm e}^{6 x}}{6} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-18+216 \,{\mathrm e}^{-6 x} c_{1} +108 x \right )^{\frac {1}{3}}}{6} \\ y \left (x \right ) &= -\frac {\left (-18+216 \,{\mathrm e}^{-6 x} c_{1} +108 x \right )^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{12} \\ y \left (x \right ) &= \frac {\left (-18+216 \,{\mathrm e}^{-6 x} c_{1} +108 x \right )^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{12} \\ \end{align*}

Problem 4975

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

Problem 4976

ODE	\[ \boxed {x^{\prime }+k x=\alpha -\beta \cos \left (\frac {\pi t}{12}\right )} \] With initial conditions \begin {align*} [x \left (0\right ) = x_{0}] \end {align*}
program solution	\[ x = \frac {-144 \,{\mathrm e}^{-k t} \cos \left (\frac {\pi t}{12}\right ) {\mathrm e}^{k t} \beta \,k^{2}-12 \,{\mathrm e}^{-k t} \pi \sin \left (\frac {\pi t}{12}\right ) {\mathrm e}^{k t} \beta k +{\mathrm e}^{-k t} \pi ^{2} {\mathrm e}^{k t} \alpha +{\mathrm e}^{-k t} \pi ^{2} k x_{0} +144 \,{\mathrm e}^{-k t} {\mathrm e}^{k t} \alpha \,k^{2}+144 \,{\mathrm e}^{-k t} k^{3} x_{0} -{\mathrm e}^{-k t} \pi ^{2} \alpha -144 \,{\mathrm e}^{-k t} \alpha \,k^{2}+144 \,{\mathrm e}^{-k t} \beta \,k^{2}}{\pi ^{2} k +144 k^{3}} \] Verified OK.
Maple solution	\[ x \left (t \right ) = \frac {-144 \cos \left (\frac {\pi t}{12}\right ) \beta \,k^{2}-12 \sin \left (\frac {\pi t}{12}\right ) \pi \beta k +\left (144 k^{3} x_{0} +144 \left (\beta -\alpha \right ) k^{2}+\pi ^{2} k x_{0} -\pi ^{2} \alpha \right ) {\mathrm e}^{-k t}+144 \alpha \,k^{2}+\pi ^{2} \alpha }{\pi ^{2} k +144 k^{3}} \]

Problem 4977

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

Problem 4978

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

Problem 4979

ODE	\[ \boxed {-2 y+y^{\prime } x=-x^{\frac {10}{3}}} \]
program solution	\[ y = -\frac {\left (3 x^{\frac {4}{3}}-4 c_{1} \right ) x^{2}}{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (3 x^{\frac {4}{3}}-4 c_{1} \right ) x^{2}}{4} \]

Problem 4980

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

Problem 4981

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

Problem 4982

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

Problem 4983

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

Problem 4984

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

Problem 4985

ODE	\[ \boxed {\theta r^{\prime }+3 r=\theta +1} \]
program solution	\[ r = \frac {3 \theta ^{4}+4 \theta ^{3}+12 c_{1}}{12 \theta ^{3}} \] Verified OK.
Maple solution	\[ r \left (\theta \right ) = \frac {\theta }{4}+\frac {1}{3}+\frac {c_{1}}{\theta ^{3}} \]

Problem 4986

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

Problem 4987

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

Problem 4988

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

Problem 4989

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

Problem 4990

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

Problem 4991

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

Problem 4992

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

Problem 4993

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

Problem 4994

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

Problem 4995

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

Problem 4996

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

Problem 4997

ODE	\[ \boxed {-4 y+y^{\prime }=32 x^{2}} \]
program solution	\[ y = -\left (8 x^{2} {\mathrm e}^{-4 x}+4 x \,{\mathrm e}^{-4 x}+{\mathrm e}^{-4 x}-c_{1} \right ) {\mathrm e}^{4 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -8 x^{2}-4 x -1+{\mathrm e}^{4 x} c_{1} \]

Problem 4998

ODE	\[ \boxed {\left (x^{2}-\frac {2}{y^{3}}\right ) y^{\prime }+2 y x=3 x^{2}} \]
program solution	\[ -x^{2} \left (x -y\right )+\frac {1}{y^{2}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\frac {\left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {1}{3}}}{2}+\frac {2 \left (-x^{3}+c_{1} \right )^{2}}{\left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {1}{3}}}+x^{3}-c_{1}}{3 x^{2}} \\ y \left (x \right ) &= \frac {\frac {\left (-i \sqrt {3}-1\right ) \left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {2}{3}}}{4}+\left (\left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {1}{3}}+\left (i \sqrt {3}-1\right ) \left (x^{3}-c_{1} \right )\right ) \left (x^{3}-c_{1} \right )}{3 \left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {1}{3}} x^{2}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {2}{3}}}{4}+\left (x^{3}-c_{1} \right ) \left (\left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {1}{3}}+\left (-i \sqrt {3}-1\right ) \left (x^{3}-c_{1} \right )\right )}{3 \left (8 x^{9}-24 c_{1} x^{6}+24 c_{1}^{2} x^{3}+12 \sqrt {3}\, \sqrt {-4 x^{9}+12 c_{1} x^{6}-12 c_{1}^{2} x^{3}+27 x^{4}+4 c_{1}^{3}}\, x^{2}-108 x^{4}-8 c_{1}^{3}\right )^{\frac {1}{3}} x^{2}} \\ \end{align*}

Problem 4999

ODE	\[ \boxed {y^{\prime }+\frac {3 y}{x}=x^{2}-4 x +3} \]
program solution	\[ y = \frac {10 x^{6}-48 x^{5}+45 x^{4}+60 c_{1}}{60 x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{3}}{6}-\frac {4 x^{2}}{5}+\frac {3 x}{4}+\frac {c_{1}}{x^{3}} \]

Problem 5000

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