2.17.21 Problems 2001 to 2100

Problem 2001

ODE	\[ \boxed {2 y-\left (y^{4}+x \right ) y^{\prime }=0} \]
program solution	\[ \frac {2 x}{\sqrt {y}}-\frac {2 y^{\frac {7}{2}}}{7} = c_{1} \] Verified OK.
Maple solution	\[ x -\frac {y \left (x \right )^{4}}{7}-\sqrt {y \left (x \right )}\, c_{1} = 0 \]

Problem 2002

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

Problem 2003

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

Problem 2004

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

Problem 2005

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

Problem 2006

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

Problem 2007

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

Problem 2008

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

Problem 2009

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

Problem 2010

ODE	\[ \boxed {\left (x^{2}-1\right ) y^{\prime }+4 y=-\left (x^{2}-1\right )^{2}} \] With initial conditions \begin {align*} [y \left (0\right ) = -6] \end {align*}
program solution	\[ y = \frac {-x^{5}+4 x^{4}+24 \ln \left (x +1\right ) x^{2}-10 x^{3}+48 \ln \left (x +1\right ) x -54 x^{2}+24 \ln \left (x +1\right )-57 x -18}{3 x^{2}-6 x +3} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-\frac {x^{3}}{3}+2 x^{2}-7 x +8 \ln \left (x +1\right )-6\right ) \left (x +1\right )^{4}}{\left (x^{2}-1\right )^{2}} \]

Problem 2011

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

Problem 2012

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

Problem 2013

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

Problem 2014

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

Problem 2015

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

Problem 2016

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

Problem 2017

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

Problem 2018

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

Problem 2019

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

Problem 2020

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

Problem 2021

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

Problem 2022

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

Problem 2023

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

Problem 2024

ODE	\[ \boxed {3 y^{\prime }+\frac {2 y}{x +1}-\frac {x}{y^{2}}=0} \]
program solution	\[ -\frac {x^{4}}{4}-\frac {2 x^{3}}{3}+\frac {\left (2 y^{3}-1\right ) x^{2}}{2}+2 x y^{3}+y^{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {{\left (\left (3 x^{4}+8 x^{3}+6 x^{2}+12 c_{1} \right ) \left (x +1\right )^{4}\right )}^{\frac {1}{3}} 18^{\frac {1}{3}}}{6 \left (x +1\right )^{2}} \\ y \left (x \right ) &= -\frac {18^{\frac {1}{3}} {\left (\left (3 x^{4}+8 x^{3}+6 x^{2}+12 c_{1} \right ) \left (x +1\right )^{4}\right )}^{\frac {1}{3}} \left (1+i \sqrt {3}\right )}{12 \left (x +1\right )^{2}} \\ y \left (x \right ) &= \frac {18^{\frac {1}{3}} {\left (\left (3 x^{4}+8 x^{3}+6 x^{2}+12 c_{1} \right ) \left (x +1\right )^{4}\right )}^{\frac {1}{3}} \left (i \sqrt {3}-1\right )}{12 \left (x +1\right )^{2}} \\ \end{align*}

Problem 2025

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

Problem 2026

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

Problem 2027

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

Problem 2028

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

Problem 2029

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

Problem 2030

ODE	\[ \boxed {2 y-\left (x^{2} y^{4}+x \right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
program solution	\[ \frac {2 \ln \left (x y^{4}+9\right )}{9}+\frac {\ln \left (y\right )}{9} = \frac {2 \ln \left (x \right )}{9}+\frac {2 \ln \left (2\right )}{9}+\frac {2 \ln \left (5\right )}{9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {100 x^{2}}{\operatorname {RootOf}\left (\textit {\_Z}^{9}-100000000 x^{9}-9 \textit {\_Z}^{8}\right )^{2}} \]

Problem 2031

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

Problem 2032

ODE	\[ \boxed {\left (1-x^{2}\right ) y^{\prime }+y x -x \left (1-x^{2}\right ) \sqrt {y}=0} \] With initial conditions \begin {align*} [y \left (0\right ) = 1] \end {align*}
program solution	\[ \sqrt {y} = \frac {x^{2}}{3}-\frac {1}{3}+\left (\frac {2}{3}-\frac {2 i}{3}\right ) \sqrt {2}\, \left (x -1\right )^{\frac {1}{4}} \left (x +1\right )^{\frac {1}{4}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\frac {4}{9}-\frac {4 i}{9}\right ) \left (x +1\right )^{\frac {5}{4}} \sqrt {2}\, \left (x -1\right )^{\frac {5}{4}}+\frac {x^{4}}{9}-\frac {16 i \sqrt {x -1}\, \sqrt {x +1}}{9}-\frac {2 x^{2}}{9}+\frac {1}{9} \]

Problem 2033

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

Problem 2034

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

Problem 2035

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

Problem 2036

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

Problem 2037

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

Problem 2038

ODE	\[ \boxed {2 y x -2 x y^{3}+\left (x^{2}+y^{2}-3 x^{2} y^{2}\right ) y^{\prime }=-x^{3}} \]
program solution	\[ \frac {\left (-2 y^{3}+x^{2}+2 y\right )^{2}}{4}-y^{6}+2 y^{4}+\frac {y^{3}}{3}-y^{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {12 x^{4}-4 x^{2}+{\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_{1} -64\right ) x^{6}-72 c_{1} x^{4}+432 c_{1}^{2} x^{2}-144 c_{1}^{2}}{3 x^{2}-1}}+12 c_{1} \right ) \left (3 x^{2}-1\right )^{2}\right )}^{\frac {2}{3}}}{{\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_{1} -64\right ) x^{6}-72 c_{1} x^{4}+432 c_{1}^{2} x^{2}-144 c_{1}^{2}}{3 x^{2}-1}}+12 c_{1} \right ) \left (3 x^{2}-1\right )^{2}\right )}^{\frac {1}{3}} \left (6 x^{2}-2\right )} \\ y \left (x \right ) &= \frac {\left (-1-i \sqrt {3}\right ) {\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_{1} -64\right ) x^{6}-72 c_{1} x^{4}+432 c_{1}^{2} x^{2}-144 c_{1}^{2}}{3 x^{2}-1}}+12 c_{1} \right ) \left (3 x^{2}-1\right )^{2}\right )}^{\frac {1}{3}}+\frac {12 \left (x^{2}-\frac {1}{3}\right ) x^{2} \left (i \sqrt {3}-1\right )}{{\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_{1} -64\right ) x^{6}-72 c_{1} x^{4}+432 c_{1}^{2} x^{2}-144 c_{1}^{2}}{3 x^{2}-1}}+12 c_{1} \right ) \left (3 x^{2}-1\right )^{2}\right )}^{\frac {1}{3}}}}{12 x^{2}-4} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) {\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_{1} -64\right ) x^{6}-72 c_{1} x^{4}+432 c_{1}^{2} x^{2}-144 c_{1}^{2}}{3 x^{2}-1}}+12 c_{1} \right ) \left (3 x^{2}-1\right )^{2}\right )}^{\frac {2}{3}}}{4}+3 \left (-1-i \sqrt {3}\right ) \left (x^{2}-\frac {1}{3}\right ) x^{2}}{{\left (\left (3 x^{4}+\sqrt {\frac {27 x^{10}-9 x^{8}+\left (216 c_{1} -64\right ) x^{6}-72 c_{1} x^{4}+432 c_{1}^{2} x^{2}-144 c_{1}^{2}}{3 x^{2}-1}}+12 c_{1} \right ) \left (3 x^{2}-1\right )^{2}\right )}^{\frac {1}{3}} \left (3 x^{2}-1\right )} \\ \end{align*}

Problem 2039

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

Problem 2040

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

Problem 2041

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

Problem 2042

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

Problem 2043

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

Problem 2044

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

Problem 2045

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

Problem 2046

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

Problem 2047

ODE	\[ \boxed {2 y x +y^{4}+\left (x y^{3}-2 x^{2}\right ) y^{\prime }=0} \]
program solution	\[ \frac {\left (y^{3}+x \right ) x}{y^{2}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\frac {\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}}{2}+\frac {2 c_{1}^{2}}{\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}}+c_{1}}{3 x} \\ y \left (x \right ) &= \frac {\left (-1-i \sqrt {3}\right ) \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {2}{3}}+4 c_{1} \left (i \sqrt {3}\, c_{1} -c_{1} +\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}\right )}{12 \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {2}{3}}+4 \left (-i \sqrt {3}\, c_{1} -c_{1} +\left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}}\right ) c_{1}}{12 \left (-108 x^{4}+12 \sqrt {81 x^{4}-12 c_{1}^{3}}\, x^{2}+8 c_{1}^{3}\right )^{\frac {1}{3}} x} \\ \end{align*}

Problem 2048

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

Problem 2049

ODE	\[ \boxed {r^{\prime }-r \cot \left (\theta \right )=0} \]
program solution	\[ r = {\mathrm e}^{c_{1}} \sin \left (\theta \right ) \] Verified OK.
Maple solution	\[ r \left (\theta \right ) = c_{1} \sin \left (\theta \right ) \]

Problem 2050

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

Problem 2051

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

Problem 2052

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

Problem 2053

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

Problem 2054

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

Problem 2055

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

Problem 2056

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

Problem 2057

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

Problem 2058

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

Problem 2059

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

Problem 2060

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

Problem 2061

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

Problem 2062

ODE	\[ \boxed {3 \,{\mathrm e}^{x} \tan \left (y\right )-\left (1-{\mathrm e}^{x}\right ) \sec \left (y\right )^{2} y^{\prime }=0} \]
program solution	\[ -\ln \left ({\mathrm e}^{x}-1\right )-\frac {\ln \left (\tan \left (y\right )\right )}{3} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\arctan \left (\frac {2 c_{1} \left ({\mathrm e}^{3 x}-3 \,{\mathrm e}^{2 x}+3 \,{\mathrm e}^{x}-1\right )}{{\mathrm e}^{6 x}-6 \,{\mathrm e}^{5 x}+15 \,{\mathrm e}^{4 x}-20 \,{\mathrm e}^{3 x}+15 \,{\mathrm e}^{2 x}+c_{1}^{2}-6 \,{\mathrm e}^{x}+1}, \frac {{\mathrm e}^{6 x}-6 \,{\mathrm e}^{5 x}+15 \,{\mathrm e}^{4 x}-20 \,{\mathrm e}^{3 x}+15 \,{\mathrm e}^{2 x}-c_{1}^{2}-6 \,{\mathrm e}^{x}+1}{{\mathrm e}^{6 x}-6 \,{\mathrm e}^{5 x}+15 \,{\mathrm e}^{4 x}-20 \,{\mathrm e}^{3 x}+15 \,{\mathrm e}^{2 x}+c_{1}^{2}-6 \,{\mathrm e}^{x}+1}\right )}{2} \]

Problem 2063

ODE	\[ \boxed {\sec \left (y\right )^{2} y^{\prime }-\tan \left (y\right )=2 x \,{\mathrm e}^{x}} \]
program solution	\[ -x^{2}+{\mathrm e}^{-x} \tan \left (y\right ) = c_{1} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 2064

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

Problem 2065

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

Problem 2066

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

Problem 2067

ODE	\[ \boxed {\left (2 x +3 y+2\right ) y^{\prime }=-x} \]
program solution	\[ \frac {\ln \left (3 x^{2}+\left (6 y+4\right ) x +9 \left (y+\frac {2}{3}\right )^{2}\right )}{3}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (3 y+2+x \right ) \sqrt {2}}{2 x}\right )}{3} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {2}{3}+\frac {\sqrt {2}\, x \tan \left (\operatorname {RootOf}\left (\sqrt {2}\, \ln \left (3\right )+\sqrt {2}\, \ln \left (\sec \left (\textit {\_Z} \right )^{2} x^{2}\right )+\sqrt {2}\, \ln \left (2\right )+2 \sqrt {2}\, c_{1} +2 \textit {\_Z} \right )\right )}{3}-\frac {x}{3} \]

Problem 2068

ODE	\[ \boxed {y^{\prime } x -5 y-\sqrt {y}\, x=0} \]
program solution	\[ \sqrt {y} = -\frac {x}{3}+c_{1} x^{\frac {5}{2}} \] Verified OK.
Maple solution	\[ \sqrt {y \left (x \right )}+\frac {x}{3}-x^{\frac {5}{2}} c_{1} = 0 \]

Problem 2069

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

Problem 2070

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

Problem 2071

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

Problem 2072

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

Problem 2073

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

Problem 2074

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

Problem 2075

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

Problem 2076

ODE	\[ \boxed {y+y^{\prime } x -y^{6} x^{3}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
program solution	\[ \frac {5 x^{3} y^{5}-2}{10 x^{5} y^{5}} = {\frac {3}{10}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-\sqrt {5}-1+i \sqrt {10-2 \sqrt {5}}\right ) 2^{\frac {1}{5}} \left (-x^{2} \left (3 x^{2}-5\right )^{4}\right )^{\frac {1}{5}}}{12 x^{3}-20 x} \]

Problem 2077

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

Problem 2078

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

Problem 2079

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

Problem 2080

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

Problem 2081

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

Problem 2082

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

Problem 2083

ODE	\[ \boxed {y^{2}+\left (x^{3}-2 y x \right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (2\right ) = 1] \end {align*}
program solution	\[ \frac {\ln \left (-2 x^{2}+3 y\right )}{6}+\frac {\ln \left (y\right )}{2} = \frac {\ln \left (x \right )}{3}-\frac {\ln \left (2\right )}{3}+\frac {\ln \left (5\right )}{6}+\frac {i \pi }{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{2} \left (-\frac {\sqrt {10}\, {\left (\sqrt {\frac {\left (20 x^{3}+20 \sqrt {x^{6}-20}\right )^{\frac {2}{3}}+20}{x \left (20 x^{3}+20 \sqrt {x^{6}-20}\right )^{\frac {1}{3}}}}+\sqrt {\frac {4 \sqrt {10}\, x \left (20 x^{3}+20 \sqrt {x^{6}-20}\right )^{\frac {1}{3}}-\sqrt {\frac {\left (20 x^{3}+20 \sqrt {x^{6}-20}\right )^{\frac {2}{3}}+20}{x \left (20 x^{3}+20 \sqrt {x^{6}-20}\right )^{\frac {1}{3}}}}\, \left (20 x^{3}+20 \sqrt {x^{6}-20}\right )^{\frac {2}{3}}-20 \sqrt {\frac {\left (20 x^{3}+20 \sqrt {x^{6}-20}\right )^{\frac {2}{3}}+20}{x \left (20 x^{3}+20 \sqrt {x^{6}-20}\right )^{\frac {1}{3}}}}}{x \left (20 x^{3}+20 \sqrt {x^{6}-20}\right )^{\frac {1}{3}} \sqrt {\frac {\left (20 x^{3}+20 \sqrt {x^{6}-20}\right )^{\frac {2}{3}}+20}{x \left (20 x^{3}+20 \sqrt {x^{6}-20}\right )^{\frac {1}{3}}}}}}\right )}^{3}}{20}+8\right )}{12} \]

Problem 2084

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

Problem 2085

ODE	\[ \boxed {y^{3}+2 x^{2} y+\left (-3 x^{3}-2 x y^{2}\right ) y^{\prime }=0} \] With initial conditions \begin {align*} [y \left (1\right ) = 1] \end {align*}
program solution	\[ -\frac {\ln \left (x^{2}+y^{2}\right )}{2}+3 \ln \left (y\right ) = \ln \left (x \right )-\frac {\ln \left (2\right )}{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {\left (54 x^{4}+6 \sqrt {3}\, \sqrt {27 x^{8}-2 x^{6}}\right )^{\frac {2}{3}}+6 x^{2}}{\left (54 x^{4}+6 \sqrt {3}\, \sqrt {27 x^{8}-2 x^{6}}\right )^{\frac {1}{3}}}}}{6} \]

Problem 2086

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

Problem 2087

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

Problem 2088

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

Problem 2089

ODE	\[ \boxed {y^{\prime \prime }+7 y^{\prime }+12 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-4 x}+c_{2} {\mathrm e}^{-3 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{-3 x}+c_{2} {\mathrm e}^{-4 x} \]

Problem 2090

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

Problem 2091

ODE	\[ \boxed {y^{\prime \prime }-7 y^{\prime }+6 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x}+\frac {c_{2} {\mathrm e}^{6 x}}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{6 x}+c_{2} {\mathrm e}^{x} \]

Problem 2092

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

Problem 2093

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

Problem 2094

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

Problem 2095

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

Problem 2096

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

Problem 2097

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

Problem 2098

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

Problem 2099

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

Problem 2100

ODE	\[ \boxed {y^{\prime \prime \prime \prime }-6 y^{\prime \prime }+8 y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-2 x}+c_{2} {\mathrm e}^{2 x}+{\mathrm e}^{\sqrt {2}\, x} c_{3} +{\mathrm e}^{-\sqrt {2}\, x} c_{4} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{2 x}+{\mathrm e}^{-2 x} c_{2} +c_{3} {\mathrm e}^{\sqrt {2}\, x}+c_{4} {\mathrm e}^{-\sqrt {2}\, x} \]