2.17.40 Problems 3901 to 4000

Problem 3901

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

Problem 3902

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

Problem 3903

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

Problem 3904

ODE	\[ \boxed {x \left (x^{3}-3 y x^{3}+4 y^{2}\right ) y^{\prime }-6 y^{3}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\operatorname {RootOf}\left (-3 x^{3} {\mathrm e}^{\textit {\_Z}}+6 c_{1} x^{3}+x^{3} \textit {\_Z} +2 \,{\mathrm e}^{2 \textit {\_Z}}\right )} \]

Problem 3905

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

Problem 3906

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

Problem 3907

ODE	\[ \boxed {x \left (x^{2}-6 y^{2}\right ) y^{\prime }-4 \left (3 y^{2}+x^{2}\right ) y=0} \]
program solution	\[ -\frac {\left (3 y^{2}+x^{2}\right )^{2}}{y}+9 y^{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {c_{1} \left (-1+\sqrt {\frac {-24 x^{6}+c_{1}^{2}}{c_{1}^{2}}}\right )}{12 x^{2}} \\ y \left (x \right ) &= \frac {c_{1} \left (1+\sqrt {\frac {-24 x^{6}+c_{1}^{2}}{c_{1}^{2}}}\right )}{12 x^{2}} \\ \end{align*}

Problem 3908

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

Problem 3909

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

Problem 3910

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

Problem 3911

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

Problem 3912

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

Problem 3913

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

Problem 3914

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

Problem 3915

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

Problem 3916

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

Problem 3917

ODE	\[ \boxed {\left (1-x^{3}+6 x^{2} y^{2}\right ) y^{\prime }-\left (6+3 y x -4 y^{3}\right ) x=0} \]
program solution	\[ 2 y^{3} x^{2}-y x^{3}-3 x^{2}+y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {6 x^{3}+\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {2}{3}}-6}{6 x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {6 i \sqrt {3}\, x^{3}-i \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {2}{3}} \sqrt {3}-6 x^{3}-6 i \sqrt {3}-\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {2}{3}}+6}{12 x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {6 i \sqrt {3}\, x^{3}-i \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {2}{3}} \sqrt {3}+6 x^{3}-6 i \sqrt {3}+\left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {2}{3}}-6}{12 x \left (162 x^{3}+6 \sqrt {3}\, \sqrt {-2 x^{9}+249 x^{6}-162 c_{1} x^{4}+27 c_{1}^{2} x^{2}-6 x^{3}+2}-54 c_{1} x \right )^{\frac {1}{3}}} \\ \end{align*}

Problem 3918

ODE	\[ \boxed {x \left (3+5 x -12 x y^{2}+4 y x^{2}\right ) y^{\prime }+\left (3+10 x -8 x y^{2}+6 y x^{2}\right ) y=0} \]
program solution	\[ x y \left (2 y x^{2}-4 x y^{2}+5 x +3\right ) = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {2^{\frac {2}{3}} {\left (\left (2 x^{5}+45 x^{3}+3 \sqrt {3}\, \sqrt {8 c_{1} x^{5}-25 x^{6}-30 x^{5}+180 c_{1} x^{3}-509 x^{4}+108 c_{1} x^{2}-900 x^{3}+108 c_{1}^{2}-540 x^{2}-108 x}+27 x^{2}+54 c_{1} \right ) x \right )}^{\frac {1}{3}}}{12 x}+\frac {\left (x^{3}+15 x +9\right ) 2^{\frac {1}{3}}}{6 {\left (\left (2 x^{5}+45 x^{3}+3 \sqrt {3}\, \sqrt {8 c_{1} x^{5}-25 x^{6}-30 x^{5}+180 c_{1} x^{3}-509 x^{4}+108 c_{1} x^{2}-900 x^{3}+108 c_{1}^{2}-540 x^{2}-108 x}+27 x^{2}+54 c_{1} \right ) x \right )}^{\frac {1}{3}}}+\frac {x}{6} \\ y \left (x \right ) &= -\frac {-4 x^{2} \left (2 x^{6}+45 x^{4}+27 x^{3}+3 x \sqrt {3}\, \sqrt {-25 x^{6}+2 \left (-15+4 c_{1} \right ) x^{5}-509 x^{4}+180 \left (c_{1} -5\right ) x^{3}+108 \left (c_{1} -5\right ) x^{2}-108 x +108 c_{1}^{2}}+54 c_{1} x \right )^{\frac {1}{3}}-2 \left (x^{3}+15 x +9\right ) x \left (i \sqrt {3}-1\right ) 2^{\frac {1}{3}}+2^{\frac {2}{3}} \left (1+i \sqrt {3}\right ) \left (2 x^{6}+45 x^{4}+27 x^{3}+3 x \sqrt {3}\, \sqrt {-25 x^{6}+2 \left (-15+4 c_{1} \right ) x^{5}-509 x^{4}+180 \left (c_{1} -5\right ) x^{3}+108 \left (c_{1} -5\right ) x^{2}-108 x +108 c_{1}^{2}}+54 c_{1} x \right )^{\frac {2}{3}}}{24 \left (2 x^{6}+45 x^{4}+27 x^{3}+3 x \sqrt {3}\, \sqrt {-25 x^{6}+2 \left (-15+4 c_{1} \right ) x^{5}-509 x^{4}+180 \left (c_{1} -5\right ) x^{3}+108 \left (c_{1} -5\right ) x^{2}-108 x +108 c_{1}^{2}}+54 c_{1} x \right )^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {4 x^{2} \left (2 x^{6}+45 x^{4}+27 x^{3}+3 x \sqrt {3}\, \sqrt {-25 x^{6}+2 \left (-15+4 c_{1} \right ) x^{5}-509 x^{4}+180 \left (c_{1} -5\right ) x^{3}+108 \left (c_{1} -5\right ) x^{2}-108 x +108 c_{1}^{2}}+54 c_{1} x \right )^{\frac {1}{3}}-2 \left (x^{3}+15 x +9\right ) x \left (1+i \sqrt {3}\right ) 2^{\frac {1}{3}}+2^{\frac {2}{3}} \left (i \sqrt {3}-1\right ) \left (2 x^{6}+45 x^{4}+27 x^{3}+3 x \sqrt {3}\, \sqrt {-25 x^{6}+2 \left (-15+4 c_{1} \right ) x^{5}-509 x^{4}+180 \left (c_{1} -5\right ) x^{3}+108 \left (c_{1} -5\right ) x^{2}-108 x +108 c_{1}^{2}}+54 c_{1} x \right )^{\frac {2}{3}}}{24 \left (2 x^{6}+45 x^{4}+27 x^{3}+3 x \sqrt {3}\, \sqrt {-25 x^{6}+2 \left (-15+4 c_{1} \right ) x^{5}-509 x^{4}+180 \left (c_{1} -5\right ) x^{3}+108 \left (c_{1} -5\right ) x^{2}-108 x +108 c_{1}^{2}}+54 c_{1} x \right )^{\frac {1}{3}} x} \\ \end{align*}

Problem 3919

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

Problem 3920

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

Problem 3921

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

Problem 3922

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

Problem 3923

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

Problem 3924

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

Problem 3925

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

Problem 3926

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

Problem 3927

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

Problem 3928

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

Problem 3929

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

Problem 3930

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

Problem 3931

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

Problem 3932

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

Problem 3933

ODE	\[ \boxed {\left (5 x^{2}+2 y^{2}\right ) y y^{\prime }+x \left (x^{2}+5 y^{2}\right )=0} \]
program solution	\[ \frac {\left (x^{2}+5 y^{2}\right )^{2}}{4}-\frac {23 y^{4}}{4} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {-10 c_{1} x^{2}-2 \sqrt {23 c_{1}^{2} x^{4}+2}}}{2 \sqrt {c_{1}}} \\ y \left (x \right ) &= \frac {\sqrt {-10 c_{1} x^{2}-2 \sqrt {23 c_{1}^{2} x^{4}+2}}}{2 \sqrt {c_{1}}} \\ y \left (x \right ) &= -\frac {\sqrt {-10 c_{1} x^{2}+2 \sqrt {23 c_{1}^{2} x^{4}+2}}}{2 \sqrt {c_{1}}} \\ y \left (x \right ) &= \frac {\sqrt {-10 c_{1} x^{2}+2 \sqrt {23 c_{1}^{2} x^{4}+2}}}{2 \sqrt {c_{1}}} \\ \end{align*}

Problem 3934

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

Problem 3935

ODE	\[ \boxed {\left (3 x^{3}+6 y x^{2}-3 x y^{2}+20 y^{3}\right ) y^{\prime }+9 y x^{2}+6 x y^{2}-y^{3}=-4 x^{3}} \]
program solution	\[ 5 y^{4}-x y^{3}+3 x^{2} y^{2}+3 y x^{3}+x^{4} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (c_{1}^{4} x^{4}+3 \textit {\_Z} \,c_{1}^{3} x^{3}+3 \textit {\_Z}^{2} c_{1}^{2} x^{2}-\textit {\_Z}^{3} c_{1} x +5 \textit {\_Z}^{4}-1\right )}{c_{1}} \]

Problem 3936

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

Problem 3937

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

Problem 3938

ODE	\[ \boxed {x \left (x -y^{3}\right ) y^{\prime }-\left (3 x +y^{3}\right ) y=0} \]
program solution	\[ -\frac {x^{2} \left (y^{3}+2 x \right )}{2 y} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {2}{3}}+3 \,{\mathrm e}^{\frac {8 c_{1}}{3}}}{3 x \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {-i \sqrt {3}\, \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {2}{3}}+3 i \sqrt {3}\, {\mathrm e}^{\frac {8 c_{1}}{3}}-\left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {2}{3}}-3 \,{\mathrm e}^{\frac {8 c_{1}}{3}}}{6 x \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {-i \sqrt {3}\, \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {2}{3}}+3 i \sqrt {3}\, {\mathrm e}^{\frac {8 c_{1}}{3}}+\left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {2}{3}}+3 \,{\mathrm e}^{\frac {8 c_{1}}{3}}}{6 x \left (-27 x^{4}+3 \sqrt {81 x^{8}-3 \,{\mathrm e}^{8 c_{1}}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 3939

ODE	\[ \boxed {x \left (2 x^{3}+y^{3}\right ) y^{\prime }-\left (2 x^{3}-y x^{2}+y^{3}\right ) y=0} \]
program solution	\[ \frac {y^{3}-4 x^{3}}{2 x^{2} y} = c_{1} -\ln \left (x \right ) \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\left (-\left (54+6 \sqrt {6 c_{1}^{3}+18 c_{1}^{2} \ln \left (x \right )+18 c_{1} \ln \left (x \right )^{2}+6 \ln \left (x \right )^{3}+81}\right )^{\frac {2}{3}}+6 \ln \left (x \right )+6 c_{1} \right ) x}{3 \left (54+6 \sqrt {6 c_{1}^{3}+18 c_{1}^{2} \ln \left (x \right )+18 c_{1} \ln \left (x \right )^{2}+6 \ln \left (x \right )^{3}+81}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (\left (\frac {i \sqrt {3}}{6}+\frac {1}{6}\right ) \left (54+6 \sqrt {6 c_{1}^{3}+18 c_{1}^{2} \ln \left (x \right )+18 c_{1} \ln \left (x \right )^{2}+6 \ln \left (x \right )^{3}+81}\right )^{\frac {2}{3}}+\left (i \sqrt {3}-1\right ) \left (\ln \left (x \right )+c_{1} \right )\right ) x}{\left (54+6 \sqrt {6 c_{1}^{3}+18 c_{1}^{2} \ln \left (x \right )+18 c_{1} \ln \left (x \right )^{2}+6 \ln \left (x \right )^{3}+81}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (\frac {\left (i \sqrt {3}-1\right ) \left (54+6 \sqrt {6 c_{1}^{3}+18 c_{1}^{2} \ln \left (x \right )+18 c_{1} \ln \left (x \right )^{2}+6 \ln \left (x \right )^{3}+81}\right )^{\frac {2}{3}}}{6}+\left (\ln \left (x \right )+c_{1} \right ) \left (1+i \sqrt {3}\right )\right ) x}{\left (54+6 \sqrt {6 c_{1}^{3}+18 c_{1}^{2} \ln \left (x \right )+18 c_{1} \ln \left (x \right )^{2}+6 \ln \left (x \right )^{3}+81}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 3940

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

Problem 3941

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

Problem 3942

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

Problem 3943

ODE	\[ \boxed {x \left (x^{4}-2 y^{3}\right ) y^{\prime }+\left (2 x^{4}+y^{3}\right ) y=0} \]
program solution	\[ \frac {y \left (2 x^{4}-y^{3}\right )}{2 x^{2}} = c_{1} \] Verified OK.
Maple solution	\[ \ln \left (x \right )-c_{1} +\frac {3 \ln \left (\frac {y \left (x \right ) \left (-2 x^{4}+y \left (x \right )^{3}\right )}{x^{\frac {16}{3}}}\right )}{10} = 0 \]

Problem 3944

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

Problem 3945

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

Problem 3946

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

Problem 3947

ODE	\[ \boxed {x \left (2-x y^{2}-2 x y^{3}\right ) y^{\prime }+2 y=-1} \]
program solution	\[ \frac {\left (-2 y-1\right ) x \ln \left (2 y+1\right )-8+\left (-4 y^{3}+2 y^{2}+2 y\right ) x}{\left (8 y+4\right ) x} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -{\frac {1}{2}} \\ y \left (x \right ) &= \frac {{\mathrm e}^{\operatorname {RootOf}\left (x \,{\mathrm e}^{3 \textit {\_Z}}-4 x \,{\mathrm e}^{2 \textit {\_Z}}+8 c_{1} x \,{\mathrm e}^{\textit {\_Z}}+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}} x +3 \,{\mathrm e}^{\textit {\_Z}} x +16\right )}}{2}-\frac {1}{2} \\ \end{align*}

Problem 3948

ODE	\[ \boxed {\left (2-10 y^{3} x^{2}+3 y^{2}\right ) y^{\prime }-x \left (1+5 y^{4}\right )=0} \]
program solution	\[ -\frac {x^{2} \left (1+5 y^{4}\right )}{2}+y^{3}+2 y = c_{1} \] Verified OK.
Maple solution	\[ -\frac {5 y \left (x \right )^{4} x^{2}}{2}-\frac {x^{2}}{2}+y \left (x \right )^{3}+2 y \left (x \right )+c_{1} = 0 \]

Problem 3949

ODE	\[ \boxed {x \left (a +y^{3} b x \right ) y^{\prime }+\left (a +c \,x^{3} y\right ) y=0} \]
program solution	\[ \frac {c \,x^{3} y-2 a}{2 x y}+\frac {b y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {3^{\frac {1}{3}} \left (-x^{2} b \left (c \,x^{2}-2 c_{1} \right ) 3^{\frac {1}{3}}+{\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{\frac {2}{3}}\right )}{3 {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{\frac {1}{3}} b x} \\ y \left (x \right ) &= -\frac {\left (\left (1+i \sqrt {3}\right ) {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{\frac {2}{3}}+x^{2} \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) b \left (c \,x^{2}-2 c_{1} \right )\right ) 3^{\frac {1}{3}}}{6 {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{\frac {1}{3}} b x} \\ y \left (x \right ) &= \frac {3^{\frac {1}{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{\frac {2}{3}}+x^{2} b \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) \left (c \,x^{2}-2 c_{1} \right )\right )}{6 {\left (\left (9 a +\sqrt {\frac {3 c^{3} x^{8}-18 c^{2} c_{1} x^{6}+36 c \,c_{1}^{2} x^{4}-24 c_{1}^{3} x^{2}+81 a^{2} b}{b}}\right ) b^{2} x^{2}\right )}^{\frac {1}{3}} b x} \\ \end{align*}

Problem 3950

ODE	\[ \boxed {x \left (1-2 y^{3} x^{2}\right ) y^{\prime }+\left (1-2 y^{2} x^{3}\right ) y=0} \]
program solution	\[ \frac {-2 x y^{2}-\frac {1}{2 x^{2}}}{y^{2}}-2 y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {{\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {1}{3}}}{6 x}+\frac {\left (-2 x +c_{1} \right )^{2} x}{6 {\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {1}{3}}}-\frac {x}{3}+\frac {c_{1}}{6} \\ y \left (x \right ) &= \frac {-2 \left (-c_{1} x +2 x^{2}\right ) {\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {1}{3}}-i \left (-c_{1}^{2} x^{2}+4 c_{1} x^{3}-4 x^{4}+{\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {2}{3}}\right ) \sqrt {3}-4 x^{4}+4 c_{1} x^{3}-c_{1}^{2} x^{2}-{\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {2}{3}}}{12 {\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {1}{3}} x} \\ y \left (x \right ) &= \frac {2 \left (c_{1} x -2 x^{2}\right ) {\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {1}{3}}+i \left (-c_{1}^{2} x^{2}+4 c_{1} x^{3}-4 x^{4}+{\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {2}{3}}\right ) \sqrt {3}-4 x^{4}+4 c_{1} x^{3}-c_{1}^{2} x^{2}-{\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {2}{3}}}{12 {\left (\left (c_{1}^{3} x^{2}-6 c_{1}^{2} x^{3}+12 c_{1} x^{4}-8 x^{5}+3 \sqrt {-6 c_{1}^{3} x^{2}+36 c_{1}^{2} x^{3}-72 c_{1} x^{4}+48 x^{5}+81}-27\right ) x \right )}^{\frac {1}{3}} x} \\ \end{align*}

Problem 3951

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

Problem 3952

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

Problem 3953

ODE	\[ \boxed {\left (x^{3}-y^{4}\right ) y^{\prime }-3 y x^{2}=0} \]
program solution	\[ -\frac {x^{3}}{y}-\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (x^{9} \textit {\_Z}^{4}+3-{\mathrm e}^{\frac {9 c_{1}}{4}} \textit {\_Z} \right ) x^{3} \]

Problem 3954

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

Problem 3955

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

Problem 3956

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

Problem 3957

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

Problem 3958

ODE	\[ \boxed {\left (x +2 y+2 y^{3} x^{2}+y^{4} x \right ) y^{\prime }+\left (1+y^{4}\right ) y=0} \]
program solution
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-1+\frac {\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+4 c_{1} x^{4}+18 c_{1}^{2} x^{2}-x^{2}-4 c_{1}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}}{2}-\frac {2 \left (3 c_{1} x^{2}-1\right )}{\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+4 c_{1} x^{4}+18 c_{1}^{2} x^{2}-x^{2}-4 c_{1}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}}}{3 c_{1} x} \\ y \left (x \right ) &= \frac {i \left (4-12 c_{1} x^{2}-\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{\frac {2}{3}}\right ) \sqrt {3}+12 c_{1} x^{2}-{\left (\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}+2\right )}^{2}}{12 \left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{\frac {1}{3}} c_{1} x} \\ y \left (x \right ) &= \frac {12 i \sqrt {3}\, c_{1} x^{2}+i \left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{\frac {2}{3}} \sqrt {3}+12 c_{1} x^{2}-4 i \sqrt {3}-\left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{\frac {2}{3}}-4 \left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}-4}{12 c_{1} x \left (108 c_{1}^{3} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{2}+18 c_{1}^{2} x^{2}+\left (4 x^{4}-4\right ) c_{1} -x^{2}}\, x c_{1} +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 3959

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

Problem 3960

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

Problem 3961

ODE	\[ \boxed {\left (x^{2}-y^{5}\right ) y^{\prime }-2 y x=0} \]
program solution	\[ -\frac {x^{2}}{y}-\frac {y^{4}}{4} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (x^{8} \textit {\_Z}^{5}+4-{\mathrm e}^{\frac {8 c_{1}}{5}} \textit {\_Z} \right ) x^{2} \]

Problem 3962

ODE	\[ \boxed {x \left (x^{3}+y^{5}\right ) y^{\prime }-\left (x^{3}-y^{5}\right ) y=0} \]
program solution	\[ -\frac {x \left (-4 y^{5}+x^{3}\right )}{4 y^{4}} = c_{1} \] Verified OK.
Maple solution	\[ \ln \left (x \right )-c_{1} +\frac {5 \ln \left (\frac {4 y \left (x \right )^{5}-x^{3}}{x^{3}}\right )}{8}-\frac {5 \ln \left (\frac {y \left (x \right )}{x^{\frac {3}{5}}}\right )}{2} = 0 \]

Problem 3963

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

Problem 3964

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

Problem 3965

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

Problem 3966

ODE	\[ \boxed {f \left (x \right ) y^{m} y^{\prime }+g \left (x \right ) y^{1+m}+h \left (x \right ) y^{n}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right )} {\left (\left (n -m -1\right ) \left (\int \frac {h \left (x \right ) {\mathrm e}^{\left (-n +m +1\right ) \left (\int \frac {g \left (x \right )}{f \left (x \right )}d x \right )}}{f \left (x \right )}d x \right )+c_{1} \right )}^{\frac {1}{-n +m +1}} \]

Problem 3967

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

Problem 3968

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

Problem 3969

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

Problem 3970

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

Problem 3971

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

Problem 3972

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

Problem 3973

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

Problem 3974

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

Problem 3975

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

Problem 3976

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

Problem 3977

ODE	\[ \boxed {x \left (\sqrt {x^{2}+y^{2}}+x \right ) y^{\prime }+y \sqrt {x^{2}+y^{2}}=0} \]
program solution	\[ \ln \left (x \right ) = -\frac {\ln \left (\frac {4 y^{2}}{x^{2}}+3\right )}{12}-\frac {\ln \left (\frac {y}{x}\right )}{3}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {\frac {y^{2}}{x^{2}}+1}}\right )}{3}-\frac {\operatorname {arctanh}\left (2 \sqrt {\frac {y^{2}}{x^{2}}+1}\right )}{6}+c_{1} \] Verified OK.
Maple solution	\[ -\left (\int _{\textit {\_b}}^{x}\frac {\sqrt {\textit {\_a}^{2}+y \left (x \right )^{2}}}{\textit {\_a} \left (2 \sqrt {\textit {\_a}^{2}+y \left (x \right )^{2}}+\textit {\_a} \right )}d \textit {\_a} \right )+\int _{}^{y \left (x \right )}\frac {\textit {\_f}^{2} \left (2 \sqrt {\textit {\_f}^{2}+x^{2}}+x \right ) \left (\int _{\textit {\_b}}^{x}\frac {1}{\sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}\, \left (2 \sqrt {\textit {\_a}^{2}+\textit {\_f}^{2}}+\textit {\_a} \right )^{2}}d \textit {\_a} \right )-x -\sqrt {\textit {\_f}^{2}+x^{2}}}{\textit {\_f} \left (2 \sqrt {\textit {\_f}^{2}+x^{2}}+x \right )}d \textit {\_f} +c_{1} = 0 \]

Problem 3978

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

Problem 3979

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

Problem 3980

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

Problem 3981

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

Problem 3982

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

Problem 3983

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

Problem 3984

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

Problem 3985

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

Problem 3986

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

Problem 3987

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

Problem 3988

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

Problem 3989

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

Problem 3990

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

Problem 3991

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

Problem 3992

ODE	\[ \boxed {{y^{\prime }}^{2}-y=x^{2}} \]
program solution	\[ -\frac {\ln \left (x \sqrt {x^{2}+y}+2 y\right )}{4}-\frac {\sqrt {17}\, \operatorname {arctanh}\left (\frac {\left (4 \sqrt {x^{2}+y}+x \right ) \sqrt {17}}{17 x}\right )}{34}+\frac {\ln \left (-x \sqrt {x^{2}+y}+2 y\right )}{4}+\frac {\sqrt {17}\, \operatorname {arctanh}\left (\frac {\left (x -4 \sqrt {x^{2}+y}\right ) \sqrt {17}}{17 x}\right )}{34}+\frac {\ln \left (-x^{4}-y x^{2}+4 y^{2}\right )}{4}+\frac {\sqrt {17}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-8 y\right ) \sqrt {17}}{17 x^{2}}\right )}{34} = c_{1} \] Verified OK. \[ \frac {\ln \left (x \sqrt {x^{2}+y}+2 y\right )}{4}+\frac {\sqrt {17}\, \operatorname {arctanh}\left (\frac {\left (4 \sqrt {x^{2}+y}+x \right ) \sqrt {17}}{17 x}\right )}{34}-\frac {\ln \left (-x \sqrt {x^{2}+y}+2 y\right )}{4}-\frac {\sqrt {17}\, \operatorname {arctanh}\left (\frac {\left (x -4 \sqrt {x^{2}+y}\right ) \sqrt {17}}{17 x}\right )}{34}+\frac {\ln \left (-x^{4}-y x^{2}+4 y^{2}\right )}{4}+\frac {\sqrt {17}\, \operatorname {arctanh}\left (\frac {\left (x^{2}-8 y\right ) \sqrt {17}}{17 x^{2}}\right )}{34} = c_{1} \] Verified OK.
Maple solution	\begin{align*} -17 \ln \left (-x^{4}-x^{2} y \left (x \right )+4 y \left (x \right )^{2}\right )-17 \ln \left (-\sqrt {y \left (x \right )+x^{2}}\, x +2 y \left (x \right )\right )+17 \ln \left (\sqrt {y \left (x \right )+x^{2}}\, x +2 y \left (x \right )\right )+\left (2 \,\operatorname {arctanh}\left (\frac {\left (4 \sqrt {y \left (x \right )+x^{2}}+x \right ) \sqrt {17}}{17 x}\right )-2 \,\operatorname {arctanh}\left (\frac {\left (x -4 \sqrt {y \left (x \right )+x^{2}}\right ) \sqrt {17}}{17 x}\right )-2 \,\operatorname {arctanh}\left (\frac {\left (x^{2}-8 y \left (x \right )\right ) \sqrt {17}}{17 x^{2}}\right )\right ) \sqrt {17}-c_{1} &= 0 \\ 17 \ln \left (-x^{4}-x^{2} y \left (x \right )+4 y \left (x \right )^{2}\right )-17 \ln \left (-\sqrt {y \left (x \right )+x^{2}}\, x +2 y \left (x \right )\right )+17 \ln \left (\sqrt {y \left (x \right )+x^{2}}\, x +2 y \left (x \right )\right )+\left (2 \,\operatorname {arctanh}\left (\frac {\left (4 \sqrt {y \left (x \right )+x^{2}}+x \right ) \sqrt {17}}{17 x}\right )-2 \,\operatorname {arctanh}\left (\frac {\left (x -4 \sqrt {y \left (x \right )+x^{2}}\right ) \sqrt {17}}{17 x}\right )+2 \,\operatorname {arctanh}\left (\frac {\left (x^{2}-8 y \left (x \right )\right ) \sqrt {17}}{17 x^{2}}\right )\right ) \sqrt {17}-c_{1} &= 0 \\ \end{align*}

Problem 3993

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

Problem 3994

ODE	\[ \boxed {{y^{\prime }}^{2}-8 y=-3 x^{2}} \]
program solution	\[ -\frac {3 \ln \left (\sqrt {-3 x^{2}+8 y}+3 x \right )}{4}+\frac {3 \ln \left (\sqrt {-3 x^{2}+8 y}-3 x \right )}{4}+\frac {\ln \left (x +\sqrt {-3 x^{2}+8 y}\right )}{4}-\frac {\ln \left (\sqrt {-3 x^{2}+8 y}-x \right )}{4}+\frac {3 \ln \left (-3 x^{2}+2 y\right )}{4}-\frac {\ln \left (-x^{2}+2 y\right )}{4} = c_{1} \] Verified OK. \[ \frac {3 \ln \left (\sqrt {-3 x^{2}+8 y}+3 x \right )}{4}-\frac {3 \ln \left (\sqrt {-3 x^{2}+8 y}-3 x \right )}{4}-\frac {\ln \left (x +\sqrt {-3 x^{2}+8 y}\right )}{4}+\frac {\ln \left (\sqrt {-3 x^{2}+8 y}-x \right )}{4}+\frac {3 \ln \left (-3 x^{2}+2 y\right )}{4}-\frac {\ln \left (-x^{2}+2 y\right )}{4} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {3 x^{2}}{8}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{6}-18 x \,\textit {\_Z}^{5}+135 x^{2} \textit {\_Z}^{4}-540 x^{3} \textit {\_Z}^{3}+\left (1215 x^{4}-16 c_{1} \right ) \textit {\_Z}^{2}+\left (-1458 x^{5}+32 c_{1} x \right ) \textit {\_Z} +729 x^{6}-16 c_{1} x^{2}\right )^{2}}{8} \\ y \left (x \right ) &= \frac {3 x^{2}}{8}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 x \,\textit {\_Z}^{5}+135 x^{2} \textit {\_Z}^{4}+540 x^{3} \textit {\_Z}^{3}+\left (1215 x^{4}-16 c_{1} \right ) \textit {\_Z}^{2}+\left (1458 x^{5}-32 c_{1} x \right ) \textit {\_Z} +729 x^{6}-16 c_{1} x^{2}\right )^{2}}{8} \\ \end{align*}

Problem 3995

ODE	\[ \boxed {{y^{\prime }}^{2}+b y=-a \,x^{2}} \]
program solution	\[ \ln \left (x \right ) = -\frac {\ln \left (-\frac {2 y b}{x^{2}}+\sqrt {-\frac {y b}{x^{2}}-a}\, b \right )}{4}-\frac {b \,\operatorname {arctanh}\left (\frac {b +4 \sqrt {-\frac {y b}{x^{2}}-a}}{\sqrt {b^{2}-16 a}}\right )}{2 \sqrt {b^{2}-16 a}}+\frac {\ln \left (-\frac {2 y b}{x^{2}}-\sqrt {-\frac {y b}{x^{2}}-a}\, b \right )}{4}-\frac {b \,\operatorname {arctanh}\left (\frac {4 \sqrt {-\frac {y b}{x^{2}}-a}-b}{\sqrt {b^{2}-16 a}}\right )}{2 \sqrt {b^{2}-16 a}}-\frac {\ln \left (\frac {4 y^{2}}{x^{4}}+\frac {y b}{x^{2}}+a \right )}{4}-\frac {b \,\operatorname {arctanh}\left (\frac {\frac {8 y}{x^{2}}+b}{\sqrt {b^{2}-16 a}}\right )}{2 \sqrt {b^{2}-16 a}}+c_{1} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 3996

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

Problem 3997

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

Problem 3998

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

Problem 3999

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

Problem 4000

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