2.17.39 Problems 3801 to 3900

Problem 3801

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

Problem 3802

ODE	\[ \boxed {x \left (1+x -2 y\right ) y^{\prime }+\left (1-2 x +y\right ) y=0} \]
program solution	\[ -\frac {3 \left (1+y+x \right )}{\left (y x \right )^{\frac {1}{3}}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {3 \,5^{\frac {1}{3}} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1} -x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{40 c_{1}}+\frac {3 x 5^{\frac {2}{3}}}{40 {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}+160 c_{1} x +80 c_{1} -x}{c_{1}}}-20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}-x -1 \\ y \left (x \right ) &= \frac {\frac {3 \,5^{\frac {1}{3}} \left (-1-i \sqrt {3}\right ) {\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2} x \right )}^{\frac {2}{3}}}{80}+\frac {3 c_{1} \left (\frac {80 \left (-x -1\right ) {\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2} x \right )}^{\frac {1}{3}}}{3}+5^{\frac {2}{3}} \left (i \sqrt {3}-1\right ) x \right )}{80}}{{\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2} x \right )}^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= -\frac {3 \left (5^{\frac {1}{3}} \left (1-i \sqrt {3}\right ) {\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2} x \right )}^{\frac {2}{3}}+c_{1} \left (\frac {80 \left (x +1\right ) {\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2} x \right )}^{\frac {1}{3}}}{3}+5^{\frac {2}{3}} x \left (1+i \sqrt {3}\right )\right )\right )}{80 {\left (-20 \left (-\frac {\sqrt {5}\, \sqrt {\frac {80 \left (x +1\right )^{2} c_{1} -x}{c_{1}}}}{20}+x +1\right ) c_{1}^{2} x \right )}^{\frac {1}{3}} c_{1}} \\ \end{align*}

Problem 3803

ODE	\[ \boxed {x \left (1-x -2 y\right ) y^{\prime }+\left (1+2 x +y\right ) y=0} \]
program solution	\[ \frac {-3+3 x -3 y}{\left (y x \right )^{\frac {1}{3}}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {3 \,5^{\frac {1}{3}} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}-160 c_{1} x +80 c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{40 c_{1}}+\frac {3 x 5^{\frac {2}{3}}}{40 {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}-160 c_{1} x +80 c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}+x -1 \\ y \left (x \right ) &= \frac {\frac {3 \,5^{\frac {1}{3}} \left (-1-i \sqrt {3}\right ) {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{\frac {2}{3}}}{80}+\frac {3 c_{1} \left (\frac {80 \left (x -1\right ) {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{3}+5^{\frac {2}{3}} \left (i \sqrt {3}-1\right ) x \right )}{80}}{c_{1} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {3 \left (5^{\frac {1}{3}} \left (1-i \sqrt {3}\right ) {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{\frac {2}{3}}+c_{1} \left (\frac {80 \left (1-x \right ) {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}}}{3}+5^{\frac {2}{3}} x \left (1+i \sqrt {3}\right )\right )\right )}{80 {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{\frac {1}{3}} c_{1}} \\ \end{align*}

Problem 3804

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

Problem 3805

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

Problem 3806

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

Problem 3807

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

Problem 3808

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

Problem 3809

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

Problem 3810

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

Problem 3811

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

Problem 3812

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

Problem 3813

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

Problem 3814

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

Problem 3815

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

Problem 3816

ODE	\[ \boxed {\left (1-y x^{2}\right ) y^{\prime }+x y^{2}=1} \]
program solution
Maple solution	\begin{align*} y \left (x \right ) &= \frac {4^{\frac {2}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {2}{3}}+\left (\left (-c_{1} +80\right ) x^{7}-160 x^{4}+80 x \right ) 4^{\frac {1}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {1}{3}}+\left (c_{1}^{2}-80 c_{1} \right ) x^{8}+160 c_{1} x^{5}-80 c_{1} x^{2}}{x^{2} 4^{\frac {2}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {2}{3}}+\left (c_{1} x^{4}-4^{\frac {1}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {1}{3}}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )} \\ y \left (x \right ) &= \frac {4^{\frac {2}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {2}{3}} \left (\sqrt {3}+i\right )+\left (2 i 4^{\frac {1}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {1}{3}}+\left (i-\sqrt {3}\right ) c_{1} x \right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right ) x}{x^{2} 4^{\frac {2}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {2}{3}} \left (\sqrt {3}+i\right )+\left (2 i 4^{\frac {1}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {1}{3}}+\left (i-\sqrt {3}\right ) c_{1} x^{4}\right ) \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )} \\ y \left (x \right ) &= \frac {\left (i-\sqrt {3}\right ) 4^{\frac {2}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {2}{3}}+\left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right ) x \left (2 i 4^{\frac {1}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {1}{3}}+c_{1} x \left (\sqrt {3}+i\right )\right )}{\left (i-\sqrt {3}\right ) x^{2} 4^{\frac {2}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {2}{3}}+\left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right ) \left (2 i 4^{\frac {1}{3}} {\left (c_{1} \left (-80+\left (c_{1} -80\right ) x^{6}+160 x^{3}\right )^{2} \left (\sqrt {5}\, \sqrt {-\frac {\left (x^{3}-1\right )^{2}}{c_{1} x^{6}-80 x^{6}+160 x^{3}-80}}-\frac {1}{4}\right )\right )}^{\frac {1}{3}}+c_{1} x^{4} \left (\sqrt {3}+i\right )\right )} \\ \end{align*}

Problem 3817

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

Problem 3818

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

Problem 3819

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

Problem 3820

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

Problem 3821

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

Problem 3822

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

Problem 3823

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

Problem 3824

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

Problem 3825

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

Problem 3826

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

Problem 3827

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

Problem 3828

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

Problem 3829

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

Problem 3830

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

Problem 3831

ODE	\[ \boxed {3 x^{2} y y^{\prime }+2 x y^{2}=-1} \]
program solution	\[ \frac {3 x^{\frac {1}{3}} \left (x y^{2}+2\right )}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\sqrt {x^{\frac {10}{3}} \left (-2 x^{\frac {1}{3}}+c_{1} \right )}}{x^{\frac {7}{3}}} \\ y \left (x \right ) &= -\frac {\sqrt {x^{\frac {10}{3}} \left (-2 x^{\frac {1}{3}}+c_{1} \right )}}{x^{\frac {7}{3}}} \\ \end{align*}

Problem 3832

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

Problem 3833

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

Problem 3834

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

Problem 3835

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

Problem 3836

ODE	\[ \boxed {x \left (3+2 y x^{2}\right ) y^{\prime }+\left (4+3 y x^{2}\right ) y=0} \]
program solution	\[ -\frac {3 \ln \left (y\right )}{2}-\frac {\ln \left (y x^{2}+2\right )}{2} = 2 \ln \left (x \right )+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (x^{2} \textit {\_Z}^{8}-2 \textit {\_Z}^{2} c_{1} -c_{1} \right )^{6} x^{2}-2 c_{1}}{x^{2} c_{1}} \]

Problem 3837

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

Problem 3838

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

Problem 3839

ODE	\[ \boxed {3 x^{4} y y^{\prime }+2 y^{2} x^{3}=1} \]
program solution	\[ \frac {\frac {3 y^{2} x^{3}}{2}+\frac {3}{5}}{x^{\frac {5}{3}}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\sqrt {5}\, \sqrt {x^{\frac {17}{3}} \left (-2+5 x^{\frac {5}{3}} c_{1} \right )}}{5 x^{\frac {13}{3}}} \\ y \left (x \right ) &= \frac {\sqrt {5}\, \sqrt {x^{\frac {17}{3}} \left (-2+5 x^{\frac {5}{3}} c_{1} \right )}}{5 x^{\frac {13}{3}}} \\ \end{align*}

Problem 3840

ODE	\[ \boxed {x^{7} y y^{\prime }-5 y x^{3}=2 x^{2}+2} \]
program solution
Maple solution	\[ -\frac {\left (y \left (x \right ) \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (x^{3} y \left (x \right )+1\right )^{2}}{x^{2}}\right ) x^{3}-c_{1} x +\operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {3}{2}\right ], -\frac {\left (x^{3} y \left (x \right )+1\right )^{2}}{x^{2}}\right )\right ) \left (\frac {x^{6} y \left (x \right )^{2}+2 x^{3} y \left (x \right )+x^{2}+1}{x^{2}}\right )^{\frac {1}{4}}+2 x^{2}}{\left (\frac {x^{6} y \left (x \right )^{2}+2 x^{3} y \left (x \right )+x^{2}+1}{x^{2}}\right )^{\frac {1}{4}} x} = 0 \]

Problem 3841

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

Problem 3842

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

Problem 3843

ODE	\[ \boxed {\left (\operatorname {g0} \left (x \right )+y \operatorname {g1} \left (x \right )\right ) y^{\prime }-\operatorname {f1} \left (x \right ) y-\operatorname {f2} \left (x \right ) y^{2}-\operatorname {f3} \left (x \right ) y^{3}=\operatorname {f0} \left (x \right )} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 3844

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

Problem 3845

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

Problem 3846

ODE	\[ \boxed {\left (y^{2}+x \right ) y^{\prime }+y=b x +a} \]
program solution	\[ -\left (\frac {b x}{2}+a -y\right ) x +\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (6 b \,x^{2}+12 a x -12 c_{1} +2 \sqrt {9 b^{2} x^{4}+36 a \,x^{3} b +36 x^{2} a^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {2}{3}}-4 x}{2 \left (6 b \,x^{2}+12 a x -12 c_{1} +2 \sqrt {9 b^{2} x^{4}+36 a \,x^{3} b +36 x^{2} a^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {i \sqrt {3}\, \left (6 b \,x^{2}+12 a x -12 c_{1} +2 \sqrt {9 b^{2} x^{4}+36 a \,x^{3} b +36 x^{2} a^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {2}{3}}+4 i \sqrt {3}\, x +\left (6 b \,x^{2}+12 a x -12 c_{1} +2 \sqrt {9 b^{2} x^{4}+36 a \,x^{3} b +36 x^{2} a^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {2}{3}}-4 x}{4 \left (6 b \,x^{2}+12 a x -12 c_{1} +2 \sqrt {9 b^{2} x^{4}+36 a \,x^{3} b +36 x^{2} a^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {i \sqrt {3}\, \left (6 b \,x^{2}+12 a x -12 c_{1} +2 \sqrt {9 b^{2} x^{4}+36 a \,x^{3} b +36 x^{2} a^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {2}{3}}+4 i \sqrt {3}\, x -\left (6 b \,x^{2}+12 a x -12 c_{1} +2 \sqrt {9 b^{2} x^{4}+36 a \,x^{3} b +36 x^{2} a^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {2}{3}}+4 x}{4 \left (6 b \,x^{2}+12 a x -12 c_{1} +2 \sqrt {9 b^{2} x^{4}+36 a \,x^{3} b +36 x^{2} a^{2}-36 b c_{1} x^{2}-72 c_{1} a x +16 x^{3}+36 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 3847

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

Problem 3848

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

Problem 3849

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

Problem 3850

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

Problem 3851

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

Problem 3852

ODE	\[ \boxed {\left (x^{2}+y^{2}\right ) y^{\prime }+2 x \left (2 x +y\right )=0} \]
program solution	\[ \frac {x^{2} \left (4 x +3 y\right )}{3}+\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {2 \left (c_{1} x^{2}-\frac {\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {2}{3}}}{4}\right )}{\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}} \sqrt {c_{1}}} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}}{4 \sqrt {c_{1}}}-\frac {\sqrt {c_{1}}\, \left (i \sqrt {3}-1\right ) x^{2}}{\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1} x^{2}+i \left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {2}{3}} \sqrt {3}+4 c_{1} x^{2}-\left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {2}{3}}}{4 \left (4-16 x^{3} c_{1}^{\frac {3}{2}}+4 \sqrt {20 x^{6} c_{1}^{3}-8 x^{3} c_{1}^{\frac {3}{2}}+1}\right )^{\frac {1}{3}} \sqrt {c_{1}}} \\ \end{align*}

Problem 3853

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

Problem 3854

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

Problem 3855

ODE	\[ \boxed {\left (a^{2}+x^{2}+y^{2}\right ) y^{\prime }+2 y x=-b^{2}-x^{2}} \]
program solution	\[ b^{2} x +\frac {x^{3}}{3}+y x^{2}+y a^{2}+\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {2 \left (a^{2}+x^{2}-\frac {\left (-12 b^{2} x -4 x^{3}-12 c_{1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} c_{1} x +6 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}\right )}{\left (-12 b^{2} x -4 x^{3}-12 c_{1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} c_{1} x +6 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {\left (\frac {1}{4}+\frac {i \sqrt {3}}{4}\right ) \left (-12 b^{2} x -4 x^{3}-12 c_{1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} c_{1} x +6 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}+\left (i \sqrt {3}-1\right ) \left (a^{2}+x^{2}\right )}{\left (-12 b^{2} x -4 x^{3}-12 c_{1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} c_{1} x +6 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (-12 b^{2} x -4 x^{3}-12 c_{1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} c_{1} x +6 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {2}{3}}}{4}+\left (1+i \sqrt {3}\right ) \left (a^{2}+x^{2}\right )}{\left (-12 b^{2} x -4 x^{3}-12 c_{1} +4 \sqrt {4 a^{6}+12 a^{4} x^{2}+12 a^{2} x^{4}+9 b^{4} x^{2}+6 b^{2} x^{4}+5 x^{6}+18 b^{2} c_{1} x +6 c_{1} x^{3}+9 c_{1}^{2}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 3856

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

Problem 3857

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

Problem 3858

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

Problem 3859

ODE	\[ \boxed {y \left (y+1\right ) y^{\prime }=\left (x +1\right ) x} \]
program solution	\[ -\frac {x^{3}}{3}+\frac {y^{3}}{3}-\frac {x^{2}}{2}+\frac {y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-1+4 x^{3}+6 x^{2}+12 c_{1} +2 \sqrt {4 x^{6}+12 x^{5}+24 c_{1} x^{3}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}{2}+\frac {1}{2 \left (-1+4 x^{3}+6 x^{2}+12 c_{1} +2 \sqrt {4 x^{6}+12 x^{5}+24 c_{1} x^{3}+9 x^{4}+36 c_{1} x^{2}-2 x^{3}+36 c_{1}^{2}-3 x^{2}-6 c_{1}}\right )^{\frac {1}{3}}}-\frac {1}{2} \\ y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{\frac {2}{3}}-i \sqrt {3}+2 \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{\frac {1}{3}}+1}{4 \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{\frac {2}{3}}-i \sqrt {3}-2 \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{\frac {1}{3}}-1}{4 \left (4 x^{3}+6 x^{2}+2 \sqrt {\left (2 x^{3}+3 x^{2}+6 c_{1} \right ) \left (2 x^{3}+3 x^{2}+6 c_{1} -1\right )}+12 c_{1} -1\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 3860

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

Problem 3861

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

Problem 3862

ODE	\[ \boxed {\left (x^{3}+2 y-y^{2}\right ) y^{\prime }+3 y x^{2}=0} \]
program solution	\[ y x^{3}-\frac {y^{3}}{3}+y^{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}}{2}+\frac {2 x^{3}+2}{\left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}}+1 \\ y \left (x \right ) &= -\frac {\left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}}{4}+\frac {-x^{3}-1}{\left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}}+1+\frac {i \sqrt {3}\, \left (x^{3}-\frac {\left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {2}{3}}}{4}+1\right )}{\left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {\frac {\left (i \sqrt {3}-1\right ) \left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {2}{3}}}{4}-1+\left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}-i \left (x^{3}+1\right ) \sqrt {3}-x^{3}}{\left (12 x^{3}+12 c_{1} +8+4 \sqrt {-4 x^{9}-3 x^{6}+18 c_{1} x^{3}+9 c_{1}^{2}+12 c_{1}}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 3863

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

Problem 3864

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

Problem 3865

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

Problem 3866

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

Problem 3867

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

Problem 3868

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

Problem 3869

ODE	\[ \boxed {\left (2 x^{2}+4 y x -y^{2}\right ) y^{\prime }+4 y x +2 y^{2}=x^{2}} \]
program solution	\[ -\frac {x^{3}}{3}+2 y x^{2}+2 x y^{2}-\frac {y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\frac {\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{2}+\frac {12 x^{2} c_{1}^{2}}{\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}+2 c_{1} x}{c_{1}} \\ y \left (x \right ) &= \frac {-\frac {\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}{4}-\frac {6 x^{2} c_{1}^{2}}{\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}+2 c_{1} x -\frac {i \sqrt {3}\, \left (-24 c_{1}^{2} x^{2}+\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {2}{3}}\right )}{4 \left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}}}{c_{1}} \\ y \left (x \right ) &= -\frac {24 i \sqrt {3}\, c_{1}^{2} x^{2}-i \sqrt {3}\, \left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {2}{3}}+24 c_{1}^{2} x^{2}-8 c_{1} x \left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}}+\left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {2}{3}}}{4 \left (108 x^{3} c_{1}^{3}+4+4 \sqrt {-135 c_{1}^{6} x^{6}+54 x^{3} c_{1}^{3}+1}\right )^{\frac {1}{3}} c_{1}} \\ \end{align*}

Problem 3870

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

Problem 3871

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

Problem 3872

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

Problem 3873

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

Problem 3874

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

Problem 3875

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

Problem 3876

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

Problem 3877

ODE	\[ \boxed {\left (3 x^{2}+2 y x +4 y^{2}\right ) y^{\prime }+6 y x +y^{2}=-2 x^{2}} \]
program solution	\[ \frac {2 x^{3}}{3}+3 y x^{2}+x y^{2}+\frac {4 y^{3}}{3} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (x^{3} c_{1}^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {1}{3}}-\frac {11 x^{2} c_{1}^{2}}{\left (x^{3} c_{1}^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {1}{3}}}-c_{1} x}{4 c_{1}} \\ y \left (x \right ) &= -\frac {11 i \sqrt {3}\, c_{1}^{2} x^{2}+i \left (x^{3} c_{1}^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {2}{3}} \sqrt {3}-11 c_{1}^{2} x^{2}+2 c_{1} x \left (x^{3} c_{1}^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {1}{3}}+\left (x^{3} c_{1}^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {2}{3}}}{8 \left (x^{3} c_{1}^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {1}{3}} c_{1}} \\ y \left (x \right ) &= \frac {11 i \sqrt {3}\, c_{1}^{2} x^{2}+i \left (x^{3} c_{1}^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {2}{3}} \sqrt {3}+11 c_{1}^{2} x^{2}-2 c_{1} x \left (x^{3} c_{1}^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {1}{3}}-\left (x^{3} c_{1}^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {2}{3}}}{8 \left (x^{3} c_{1}^{3}+8+2 \sqrt {333 c_{1}^{6} x^{6}+4 x^{3} c_{1}^{3}+16}\right )^{\frac {1}{3}} c_{1}} \\ \end{align*}

Problem 3878

ODE	\[ \boxed {\left (1-3 x +2 y\right )^{2} y^{\prime }-\left (4+2 x -3 y\right )^{2}=0} \]
program solution	\[ \frac {5 \ln \left (-x -9+4 y\right )}{9}+\frac {5 \ln \left (-4 x +6+y\right )}{9}-\frac {\ln \left (-5 x +5 y-3\right )}{9} = c_{1} \] Verified OK.
Maple solution	\[ \text {Expression too large to display} \]

Problem 3879

ODE	\[ \boxed {\left (1-3 y x^{2}+6 y^{2}\right ) y^{\prime }-3 x y^{2}=-x^{2}} \]
program solution	\[ -\frac {3 x^{2} y^{2}}{2}+\frac {x^{3}}{3}+2 y^{3}+y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}-162 c_{1} x^{6}+144 x^{6}+216 x^{5}+864 c_{1} x^{3}-27 x^{4}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}{12}+\frac {3 x^{4}-8}{4 \left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}-162 c_{1} x^{6}+144 x^{6}+216 x^{5}+864 c_{1} x^{3}-27 x^{4}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}+\frac {x^{2}}{4} \\ y \left (x \right ) &= \frac {24+i \left (-24+9 x^{4}-\left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {2}{3}}\right ) \sqrt {3}-9 x^{4}+6 x^{2} \left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}-\left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {2}{3}}}{24 \left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {24+i \left (-9 x^{4}+\left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {2}{3}}+24\right ) \sqrt {3}-9 x^{4}+6 x^{2} \left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}-\left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {2}{3}}}{24 \left (-108 x^{2}-144 x^{3}-432 c_{1} +27 x^{6}+12 \sqrt {-54 x^{9}+\left (-162 c_{1} +144\right ) x^{6}+216 x^{5}-27 x^{4}+864 c_{1} x^{3}+648 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 3880

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

Problem 3881

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

Problem 3882

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

Problem 3883

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

Problem 3884

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

Problem 3885

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

Problem 3886

ODE	\[ \boxed {x \left (3 x -y^{2}\right ) y^{\prime }+\left (5 x -2 y^{2}\right ) y=0} \]
program solution	\[ \frac {3 \ln \left (y\right )}{13}-\frac {\ln \left (5 y^{2}-13 x \right )}{65} = -\frac {2 \ln \left (x \right )}{5}+c_{1} \] Verified OK.
Maple solution	\[ \ln \left (x \right )-c_{1} -\frac {2 \ln \left (\frac {5 y \left (x \right )^{2}-13 x}{x}\right )}{65}+\frac {6 \ln \left (\frac {y \left (x \right )}{\sqrt {x}}\right )}{13} = 0 \]

Problem 3887

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

Problem 3888

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

Problem 3889

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

Problem 3890

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

Problem 3891

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

Problem 3892

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

Problem 3893

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

Problem 3894

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

Problem 3895

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

Problem 3896

ODE	\[ \boxed {x \left (x^{2}-2 y^{2}\right ) y^{\prime }-\left (2 x^{2}-y^{2}\right ) y=0} \]
program solution	\[ -\frac {3 \left (x^{2}+y^{2}\right )}{2 \left (y x \right )^{\frac {2}{3}}} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {6 x^{2} c_{1} \sqrt {3}\, \sqrt {2}\, \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}}{\left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {2}{3}}+2 \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}+4\right ) \sqrt {\frac {\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {2}{3}}+2 \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}+4}{\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}}}} \\ y \left (x \right ) &= \frac {6 x^{2} c_{1} \sqrt {3}\, \sqrt {2}\, \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}}{\left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {2}{3}}+2 \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}+4\right ) \sqrt {\frac {\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {2}{3}}+2 \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}+4}{\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}}}} \\ y \left (x \right ) &= \frac {24 x^{2} c_{1} \sqrt {3}\, \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}}{\left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}-2\right ) \sqrt {\frac {\left (-\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}+2\right ) \left (-2+i \left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}+2\right ) \sqrt {3}+\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}\right )}{\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}}}\, \left (-2+i \left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}+2\right ) \sqrt {3}+\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}\right )} \\ y \left (x \right ) &= \frac {24 x^{2} c_{1} \sqrt {3}\, \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}}{\left (-\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}+2\right ) \sqrt {\frac {\left (-\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}+2\right ) \left (-2+i \left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}+2\right ) \sqrt {3}+\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}\right )}{\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}}}\, \left (-2+i \left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}+2\right ) \sqrt {3}+\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}\right )} \\ y \left (x \right ) &= -\frac {24 x^{2} c_{1} \sqrt {3}\, \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}}{\sqrt {\frac {\left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}-2\right ) \left (2+i \left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}+2\right ) \sqrt {3}-\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}\right )}{\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}}}\, \left (\left (i \sqrt {3}-1\right ) \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}+2 i \sqrt {3}+2\right ) \left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}-2\right )} \\ y \left (x \right ) &= \frac {24 x^{2} c_{1} \sqrt {3}\, \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}}{\sqrt {\frac {\left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}-2\right ) \left (2+i \left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}+2\right ) \sqrt {3}-\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}\right )}{\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}}}\, \left (\left (i \sqrt {3}-1\right ) \left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}+2 i \sqrt {3}+2\right ) \left (\left (8-108 c_{1}^{2} x^{2}+12 \sqrt {3}\, \sqrt {27 c_{1}^{4} x^{4}-4 c_{1}^{2} x^{2}}\right )^{\frac {1}{3}}-2\right )} \\ \end{align*}

Problem 3897

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

Problem 3898

ODE	\[ \boxed {2 x \left (5 x^{2}+y^{2}\right ) y^{\prime }-y x^{2}+y^{3}=0} \]
program solution	\[ \frac {10 \ln \left (y\right )}{9}-\frac {2 \ln \left (y^{2}+3 x^{2}\right )}{9} = -\frac {\ln \left (x \right )}{3}+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {RootOf}\left (\textit {\_Z}^{45} c_{1} x^{9}-\textit {\_Z}^{18}-6 \textit {\_Z}^{9}-9\right )^{\frac {9}{2}} x \]

Problem 3899

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

Problem 3900

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