2.17.151 Problems 15001 to 15100

Problem 15001

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

Problem 15002

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

Problem 15003

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

Problem 15004

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

Problem 15005

ODE	\[ \boxed {x^{2} y^{\prime }+\sin \left (2 y\right )=1} \] With initial conditions \begin {align*} \left [y \left (\infty \right ) = \frac {11 \pi }{4}\right ] \end {align*}
program solution	\[ \frac {1}{x}-\frac {1}{\tan \left (y\right )-1} = {\frac {1}{2}} \] Warning, solution could not be verified
Maple solution	\[ y = -\arctan \left (\frac {x +2}{x -2}\right )+3 \pi \]

Problem 15006

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

Problem 15007

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

Problem 15008

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

Problem 15009

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

Problem 15010

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

Problem 15011

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

Problem 15012

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

Problem 15013

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

Problem 15014

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

Problem 15015

ODE	\[ \boxed {3 y-\left (3 x -7 y-3\right ) y^{\prime }=7 x -7} \]
program solution	\[ \frac {2 \ln \left (-x +1+y\right )}{3}+\frac {5 \ln \left (x -1+y\right )}{3} = c_{1} \] Verified OK.
Maple solution	\[ \text {Expression too large to display} \]

Problem 15016

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

Problem 15017

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

Problem 15018

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

Problem 15019

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

Problem 15020

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

Problem 15021

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

Problem 15022

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

Problem 15023

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

Problem 15024

ODE	\[ \boxed {y \left (1+\sqrt {y^{4} x^{2}+1}\right )+2 y^{\prime } x=0} \]
program solution	\[ \frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {y^{4} x^{2}+1}}\right )}{2} = \frac {\ln \left (x \right )}{2}+c_{1} \] 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 15025

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

Problem 15026

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

Problem 15027

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

Problem 15028

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

Problem 15029

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

Problem 15030

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

Problem 15031

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

Problem 15032

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

Problem 15033

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

Problem 15034

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

Problem 15035

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

Problem 15036

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

Problem 15037

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

Problem 15038

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

Problem 15039

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

Problem 15040

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

Problem 15041

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

Problem 15042

ODE	\[ \boxed {\sin \left (x \right ) y^{\prime }-\cos \left (x \right ) y=-\frac {\sin \left (x \right )^{2}}{x^{2}}} \] With initial conditions \begin {align*} [y \left (\infty \right ) = 0] \end {align*}
program solution	N/A
Maple solution	\[ y \left (x \right ) = \frac {\sin \left (x \right )}{x} \]

Problem 15043

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

Problem 15044

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

Problem 15045

ODE	\[ \boxed {2 y^{\prime } x +y=\left (x^{2}+1\right ) {\mathrm e}^{x}} \] With initial conditions \begin {align*} [y \left (-\infty \right ) = 1] \end {align*}
program solution	\[ y = \frac {4 x^{\frac {3}{2}} {\mathrm e}^{x}-6 \,{\mathrm e}^{x} \sqrt {x}+7 \sqrt {\pi }\, \operatorname {erfi}\left (\sqrt {x}\right )+4 c_{1}}{8 \sqrt {x}} \] Warning, solution could not be verified
Maple solution	\[ y \left (x \right ) = \frac {\infty i}{\sqrt {\operatorname {signum}\left (x \right )}} \]

Problem 15046

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

Problem 15047

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

Problem 15048

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

Problem 15049

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

Problem 15050

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

Problem 15051

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

Problem 15052

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

Problem 15053

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

Problem 15054

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

Problem 15055

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

Problem 15056

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

Problem 15057

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

Problem 15058

ODE	\[ \boxed {y^{\prime }-\tan \left (y\right )-\frac {{\mathrm e}^{x}}{\cos \left (y\right )}=0} \]
program solution	\[ -x +\sin \left (y\right ) {\mathrm e}^{-x} = c_{1} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 15059

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

Problem 15060

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

Problem 15061

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

Problem 15062

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

Problem 15063

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

Problem 15064

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

Problem 15065

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

Problem 15066

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

Problem 15067

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

Problem 15068

ODE	\[ \boxed {\frac {\sin \left (2 x \right )}{y}+\left (y-\frac {\sin \left (x \right )^{2}}{y^{2}}\right ) y^{\prime }=-x} \]
program solution	\[ \frac {x^{2}}{2}-\frac {\cos \left (2 x \right )}{2 y}+\frac {y^{2}}{2}+\frac {1}{2 y} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 x^{2} c_{1}^{2}+96 c_{1}^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{\frac {2}{3}}-12 x^{2}-24 c_{1}}{6 \left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 x^{2} c_{1}^{2}+96 c_{1}^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{12}\right ) \left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 x^{2} c_{1}^{2}+96 c_{1}^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{\frac {2}{3}}+\left (i \sqrt {3}-1\right ) \left (x^{2}+2 c_{1} \right )}{\left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 x^{2} c_{1}^{2}+96 c_{1}^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 x^{2} c_{1}^{2}+96 c_{1}^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{\frac {2}{3}} \left (i \sqrt {3}-1\right )}{12}+\left (x^{2}+2 c_{1} \right ) \left (1+i \sqrt {3}\right )}{\left (-108+108 \cos \left (2 x \right )+12 \sqrt {12 x^{6}+72 c_{1} x^{4}+144 x^{2} c_{1}^{2}+96 c_{1}^{3}+81-162 \cos \left (2 x \right )+81 \cos \left (2 x \right )^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 15069

ODE	\[ \boxed {-y+\left (2 y-x +3 y^{2}\right ) y^{\prime }=-3 x^{2}+2 x} \]
program solution	\[ x \left (x^{2}-x -y\right )+y^{3}+y^{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-36 x -108 x^{3}+108 x^{2}-108 c_{1} -8+12 \sqrt {81 x^{6}-162 x^{5}+162 c_{1} x^{3}+135 x^{4}-162 c_{1} x^{2}-54 x^{3}+81 c_{1}^{2}+54 c_{1} x -15 x^{2}+12 c_{1}}\right )^{\frac {1}{3}}}{6}+\frac {2 x +\frac {2}{3}}{\left (-36 x -108 x^{3}+108 x^{2}-108 c_{1} -8+12 \sqrt {81 x^{6}-162 x^{5}+162 c_{1} x^{3}+135 x^{4}-162 c_{1} x^{2}-54 x^{3}+81 c_{1}^{2}+54 c_{1} x -15 x^{2}+12 c_{1}}\right )^{\frac {1}{3}}}-\frac {1}{3} \\ y \left (x \right ) &= \frac {i \left (4-\left (-36 x -108 x^{3}+108 x^{2}-108 c_{1} -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_{1} -54\right ) x^{3}+\left (-162 c_{1} -15\right ) x^{2}+54 c_{1} x +81 c_{1}^{2}+12 c_{1}}\right )^{\frac {2}{3}}+12 x \right ) \sqrt {3}-12 x -{\left (\left (-36 x -108 x^{3}+108 x^{2}-108 c_{1} -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_{1} -54\right ) x^{3}+\left (-162 c_{1} -15\right ) x^{2}+54 c_{1} x +81 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}+2\right )}^{2}}{12 \left (-36 x -108 x^{3}+108 x^{2}-108 c_{1} -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_{1} -54\right ) x^{3}+\left (-162 c_{1} -15\right ) x^{2}+54 c_{1} x +81 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \left (\left (-36 x -108 x^{3}+108 x^{2}-108 c_{1} -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_{1} -54\right ) x^{3}+\left (-162 c_{1} -15\right ) x^{2}+54 c_{1} x +81 c_{1}^{2}+12 c_{1}}\right )^{\frac {2}{3}}-12 x -4\right ) \sqrt {3}-12 x -{\left (\left (-36 x -108 x^{3}+108 x^{2}-108 c_{1} -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_{1} -54\right ) x^{3}+\left (-162 c_{1} -15\right ) x^{2}+54 c_{1} x +81 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}+2\right )}^{2}}{12 \left (-36 x -108 x^{3}+108 x^{2}-108 c_{1} -8+12 \sqrt {81 x^{6}-162 x^{5}+135 x^{4}+\left (162 c_{1} -54\right ) x^{3}+\left (-162 c_{1} -15\right ) x^{2}+54 c_{1} x +81 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 15070

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

Problem 15071

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

Problem 15072

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

Problem 15073

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

Problem 15074

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

Problem 15075

ODE	\[ \boxed {3 y x^{2}+y^{3}+\left (x^{3}+3 x y^{2}\right ) y^{\prime }=0} \]
program solution	\[ y \left (y^{2}+x^{2}\right ) x = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {12^{\frac {1}{3}} \left (x^{4} c_{1}^{2} 12^{\frac {1}{3}}-{\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{\frac {2}{3}}\right )}{6 c_{1} x {\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {3^{\frac {1}{3}} 2^{\frac {2}{3}} \left (\left (1+i \sqrt {3}\right ) {\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{\frac {2}{3}}+c_{1}^{2} 2^{\frac {2}{3}} x^{4} \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right )\right )}{12 {\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{\frac {1}{3}} x c_{1}} \\ y \left (x \right ) &= \frac {\left (\left (i \sqrt {3}-1\right ) {\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{\frac {2}{3}}+c_{1}^{2} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) 2^{\frac {2}{3}} x^{4}\right ) 3^{\frac {1}{3}} 2^{\frac {2}{3}}}{12 {\left (\left (\sqrt {3}\, \sqrt {4 c_{1}^{4} x^{8}+27}+9\right ) x^{2} c_{1} \right )}^{\frac {1}{3}} x c_{1}} \\ \end{align*}

Problem 15076

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

Problem 15077

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

Problem 15078

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

Problem 15079

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

Problem 15080

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

Problem 15081

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

Problem 15082

ODE	\[ \boxed {2 x y^{2}-3 y^{3}+\left (7-3 x y^{2}\right ) y^{\prime }=0} \]
program solution	\[ x \left (x -3 y\right )-\frac {7}{y} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {x^{2}+c_{1} +\sqrt {x^{4}+2 c_{1} x^{2}+c_{1}^{2}-84 x}}{6 x} \\ y \left (x \right ) &= \frac {x^{2}-\sqrt {x^{4}+2 c_{1} x^{2}+c_{1}^{2}-84 x}+c_{1}}{6 x} \\ \end{align*}

Problem 15083

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

Problem 15084

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

Problem 15085

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

Problem 15086

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

Problem 15087

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

Problem 15088

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

Problem 15089

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

Problem 15090

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

Problem 15091

ODE	\[ \boxed {{y^{\prime }}^{3}+\left (x +2\right ) {\mathrm e}^{y}=0} \]
program solution	\[ y = \ln \left (\frac {1728}{\left (3 \left (-x -2\right )^{\frac {4}{3}}-4 c_{1} \right )^{3}}\right ) \] Verified OK. \[ y = \ln \left (\frac {1728}{\left (3 \left (-x -2\right )^{\frac {4}{3}}-4 c_{1} \right )^{3}}\right ) \] Verified OK. \[ y = \ln \left (\frac {1728}{\left (3 \left (-x -2\right )^{\frac {4}{3}}-4 c_{1} \right )^{3}}\right ) \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= 3 \ln \left (12\right )-3 \ln \left (\left (6+3 x \right ) \left (2+x \right )^{\frac {1}{3}}+4 c_{1} \right ) \\ y \left (x \right ) &= 3 \ln \left (24\right )-3 \ln \left (-3 \left (1+i \sqrt {3}\right ) \left (2+x \right )^{\frac {4}{3}}+8 c_{1} \right ) \\ y \left (x \right ) &= 3 \ln \left (24\right )-3 \ln \left (3 \left (i \sqrt {3}-1\right ) \left (2+x \right )^{\frac {4}{3}}+8 c_{1} \right ) \\ \end{align*}

Problem 15092

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

Problem 15093

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

Problem 15094

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

Problem 15095

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

Problem 15096

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

Problem 15097

ODE	\[ \boxed {-\ln \left (y^{\prime }\right )-\sin \left (y^{\prime }\right )=-x} \]
program solution	\[ y = \int \operatorname {RootOf}\left (\textit {\_Z} -{\mathrm e}^{-\sin \left (\textit {\_Z} \right )+x}\right )d x +c_{1} \] Warning, solution could not be verified
Maple solution	\[ y \left (x \right ) = \int \operatorname {RootOf}\left (-x +\ln \left (\textit {\_Z} \right )+\sin \left (\textit {\_Z} \right )\right )d x +c_{1} \]

Problem 15098

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

Problem 15099

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

Problem 15100

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