2.17.87 Problems 8601 to 8700

Problem 8601

ODE	\[ \boxed {\left (x^{n \left (n +1\right )} y-1\right ) y^{\prime }+2 \left (n +1\right )^{2} x^{n -1} \left (x^{n^{2}} y^{2}-1\right )=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 8602

ODE	\[ \boxed {\left (y-x \right ) \sqrt {x^{2}+1}\, y^{\prime }-a \sqrt {\left (y^{2}+1\right )^{3}}=0} \]
program solution	\[ \frac {2 a \arctan \left (\frac {a \tan \left (\frac {\arctan \left (x \right )}{2}-\frac {\arctan \left (y\right )}{2}\right )+1}{\sqrt {a^{2}-1}}\right )}{\sqrt {a^{2}-1}}+\arctan \left (y\right )-c_{1} = 0 \] Warning, solution could not be verified
Maple solution	\begin{align*} y \left (x \right ) &= \frac {-x +\sqrt {-a^{2} \left (x^{2}+1\right )^{2} \left (a^{2}-1\right )}}{a^{2} x^{2}+a^{2}-1} \\ y \left (x \right ) &= \frac {-x -\sqrt {-a^{2} \left (x^{2}+1\right )^{2} \left (a^{2}-1\right )}}{a^{2} x^{2}+a^{2}-1} \\ \frac {\sqrt {2}\, \sqrt {\frac {a^{2}}{1+\cos \left (2 \arctan \left (x \right )-2 \arctan \left (y \left (x \right )\right )\right )}}\, \cos \left (\arctan \left (x \right )-\arctan \left (y \left (x \right )\right )\right ) \arctan \left (\frac {\cos \left (\arctan \left (x \right )-\arctan \left (y \left (x \right )\right )\right )}{\sqrt {a^{2}-1}}\right )+\arctan \left (\frac {\sqrt {a^{2}-1}\, \tan \left (\arctan \left (x \right )-\arctan \left (y \left (x \right )\right )\right )}{a}\right ) a -\sqrt {a^{2}-1}\, \left (c_{1} -\arctan \left (y \left (x \right )\right )\right )}{\sqrt {a^{2}-1}} &= 0 \\ \end{align*}

Problem 8603

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

Problem 8604

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

Problem 8605

ODE	\[ \boxed {\left (g_{1} \left (x \right ) y+g_{0} \left (x \right )\right ) y^{\prime }-f_{1} \left (x \right ) y-f_{2} \left (x \right ) y^{2}-f_{3} \left (x \right ) y^{3}=f_{0} \left (x \right )} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 8606

ODE	\[ \boxed {\left (y^{2}-x \right ) y^{\prime }-y=-x^{2}} \]
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 8607

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

Problem 8608

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

Problem 8609

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

Problem 8610

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

Problem 8611

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

Problem 8612

ODE	\[ \boxed {\left (y^{2}-x^{2}\right ) y^{\prime }+2 y x=0} \]
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 8613

ODE	\[ \boxed {\left (y^{2}+x^{4}\right ) y^{\prime }-4 x^{3} y=0} \]
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 8614

ODE	\[ \boxed {\left (y^{2}+4 \sin \left (x \right )\right ) y^{\prime }=\cos \left (x \right )} \]
program solution	\[ -\sin \left (x \right ) {\mathrm e}^{-4 y}-\frac {\left (8 y^{2}+4 y+1\right ) {\mathrm e}^{-4 y}}{32} = c_{1} \] Verified OK.
Maple solution	\[ \frac {\left (-8 y \left (x \right )^{2}-4 y \left (x \right )-32 \sin \left (x \right )-1\right ) {\mathrm e}^{-4 y \left (x \right )}}{32}+c_{1} = 0 \]

Problem 8615

ODE	\[ \boxed {\left (y^{2}+2 y+x \right ) y^{\prime }+\left (y+x \right )^{2} y^{2}+y \left (y+1\right )=0} \]
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 c_{1} -2 x} \\ 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 8616

ODE	\[ \boxed {\left (y+x \right )^{2} y^{\prime }=a^{2}} \]
program solution	\[ y-a \arctan \left (\frac {y+x}{a}\right ) = c_{1} \] 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 8617

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

Problem 8618

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

Problem 8619

ODE	\[ \boxed {3 \left (y^{2}-x^{2}\right ) y^{\prime }+2 y^{3}-6 x \left (x +1\right ) y=3 \,{\mathrm e}^{x}} \]
program solution	\[ \left (-3 y x^{2}+y^{3}\right ) {\mathrm e}^{2 x}-{\mathrm e}^{3 x} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {2^{\frac {1}{3}} \left (2 x^{2} {\mathrm e}^{4 x}+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 x^{2} {\mathrm e}^{4 x} \left (i \sqrt {3}-1\right )+2^{\frac {1}{3}} \left (1+i \sqrt {3}\right ) {\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 ) 2^{\frac {1}{3}} {\mathrm e}^{-2 x}}{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 {2^{\frac {1}{3}} {\mathrm e}^{-2 x} \left (-2 x^{2} {\mathrm e}^{4 x} \left (1+i \sqrt {3}\right )+2^{\frac {1}{3}} \left (i \sqrt {3}-1\right ) {\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 8620

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

Problem 8621

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

Problem 8622

ODE	\[ \boxed {\left (2 y-3 x +1\right )^{2} y^{\prime }-\left (3 y-2 x -4\right )^{2}=0} \]
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 8623

ODE	\[ \boxed {\left (2 y-4 x +1\right )^{2} y^{\prime }-\left (y-2 x \right )^{2}=0} \]
program solution	\[ x = -\frac {4 y}{7}+\frac {8 x}{7}+\frac {2 \ln \left (7 \left (y-2 x \right )^{2}+8 y-16 x +2\right )}{49}+\frac {9 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (14 y-28 x +8\right ) \sqrt {2}}{4}\right )}{98}+c_{1} \] Verified OK.
Maple solution	\[ -\frac {x}{7}-\frac {9 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (7 y \left (x \right )-14 x +4\right ) \sqrt {2}}{2}\right )}{98}-\frac {2 \ln \left (7 \left (y \left (x \right )-2 x \right )^{2}+8 y \left (x \right )-16 x +2\right )}{49}+\frac {4 y \left (x \right )}{7}-c_{1} = 0 \]

Problem 8624

ODE	\[ \boxed {\left (6 y^{2}-3 y x^{2}+1\right ) y^{\prime }-3 x y^{2}=-x} \]
program solution	\[ -\frac {x^{2} \left (3 y^{2}-1\right )}{2}+2 y^{3}+y = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\left (-324 x^{2}-432 c_{1} +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_{1} x^{6}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}{12}+\frac {3 x^{4}-8}{4 \left (-324 x^{2}-432 c_{1} +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_{1} x^{6}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}}+\frac {x^{2}}{4} \\ y \left (x \right ) &= \frac {9 i \sqrt {3}\, x^{4}-i \sqrt {3}\, \left (-324 x^{2}-432 c_{1} +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_{1} x^{6}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {2}{3}}-9 x^{4}+6 x^{2} \left (-324 x^{2}-432 c_{1} +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_{1} x^{6}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}-24 i \sqrt {3}-\left (-324 x^{2}-432 c_{1} +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_{1} x^{6}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {2}{3}}+24}{24 \left (-324 x^{2}-432 c_{1} +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_{1} x^{6}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {9 i \sqrt {3}\, x^{4}-i \sqrt {3}\, \left (-324 x^{2}-432 c_{1} +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_{1} x^{6}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {2}{3}}+9 x^{4}-6 x^{2} \left (-324 x^{2}-432 c_{1} +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_{1} x^{6}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}-24 i \sqrt {3}+\left (-324 x^{2}-432 c_{1} +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_{1} x^{6}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {2}{3}}-24}{24 \left (-324 x^{2}-432 c_{1} +27 x^{6}+12 \sqrt {-81 x^{8}-162 c_{1} x^{6}+621 x^{4}+1944 c_{1} x^{2}+1296 c_{1}^{2}+96}\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 8625

ODE	\[ \boxed {\left (6 y-x \right )^{2} y^{\prime }-6 y^{2}+2 y x=-a} \]
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 8626

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

Problem 8627

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

Problem 8628

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

Problem 8629

ODE	\[ \boxed {x \left (y^{2}-3 x \right ) y^{\prime }+2 y^{3}-5 y x=0} \]
program solution	\[ -\frac {\ln \left (5 y^{2}-13 x \right )}{65}+\frac {3 \ln \left (y\right )}{13} = -\frac {2 \ln \left (x \right )}{5}+c_{1} \] 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 8630

ODE	\[ \boxed {x \left (y^{2}+x^{2}-a \right ) y^{\prime }-\left (y^{2}+x^{2}+a \right ) y=0} \]
program solution	\[ \frac {-x^{2}+y^{2}+a}{y x} = c_{1} \] Verified OK.
Maple solution	\begin{align*} \frac {y \left (x \right )^{2} \left (-x^{2}+a \right )}{-x^{2}-y \left (x \right )^{2}+a} &= -\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 )}{-x^{2}-y \left (x \right )^{2}+a} &= \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 8631

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

Problem 8632

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

Problem 8633

ODE	\[ \boxed {2 x \left (y^{2}+5 x^{2}\right ) y^{\prime }+y^{3}-y x^{2}=0} \]
program solution	\[ -\frac {2 \ln \left (3 x^{2}+y^{2}\right )}{9}+\frac {10 \ln \left (y\right )}{9} = -\frac {\ln \left (x \right )}{3}+c_{1} \] 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 8634

ODE	\[ \boxed {3 y^{2} x y^{\prime }+y^{3}=2 x} \]
program solution	\[ y^{3} x -x^{2} = c_{1} \] 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*}

Problem 8635

ODE	\[ \boxed {\left (3 x y^{2}-x^{2}\right ) y^{\prime }+y^{3}-2 y x=0} \]
program solution	\[ -x y \left (-y^{2}+x \right ) = c_{1} \] 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}} \left (\left (-i \sqrt {3}-1\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 ) x^{3} 2^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{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}}+x^{3} 2^{\frac {2}{3}} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right )\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 8636

ODE	\[ \boxed {6 y^{2} x y^{\prime }+2 y^{3}=-x} \]
program solution	\[ 2 y^{3} x +\frac {x^{2}}{2} = c_{1} \] 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 8637

ODE	\[ \boxed {\left (6 x y^{2}+x^{2}\right ) y^{\prime }-y \left (3 y^{2}-x \right )=0} \]
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 8638

ODE	\[ \boxed {\left (y^{2} x^{2}+x \right ) y^{\prime }+y=0} \]
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 (4 c_{1} +x \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 (4 c_{1} +x \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 (4 c_{1} +x \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 (4 c_{1} +x \right )}\right )}}{2 c_{1} x} \\ \end{align*}

Problem 8639

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

Problem 8640

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

Problem 8641

ODE	\[ \boxed {\left (y^{3}-3 x \right ) y^{\prime }-3 y=-x^{2}} \]
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 8642

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

Problem 8643

ODE	\[ \boxed {\left (y^{2}+x^{2}+a \right ) y y^{\prime }+x \left (y^{2}+x^{2}-a \right )=0} \]
program solution	\[ \frac {\left (-x^{2}-y^{2}+a \right )^{2}}{4}+a y^{2} = c_{1} \] 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 8644

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

Problem 8645

ODE	\[ \boxed {\left (2 y^{3}+y\right ) y^{\prime }=2 x^{3}+x} \]
program solution	\[ -\frac {\left (2 x^{2}+1\right )^{2}}{8}+\frac {\left (2 y^{2}+1\right )^{2}}{8} = c_{1} \] 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 8646

ODE	\[ \boxed {\left (2 y^{3}+5 y x^{2}\right ) y^{\prime }+5 x y^{2}=-x^{3}} \]
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 8647

ODE	\[ \boxed {\left (20 y^{3}-3 x y^{2}+6 y x^{2}+3 x^{3}\right ) y^{\prime }-y^{3}+6 x y^{2}+9 y x^{2}=-4 x^{3}} \]
program solution	\[ 5 y^{4}-y^{3} x +3 y^{2} x^{2}+3 x^{3} y+x^{4} = c_{1} \] 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 8648

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

Problem 8649

ODE	\[ \boxed {\left (2 a y^{3}+3 a y^{2} x -b \,x^{3}+c \,x^{2}\right ) y^{\prime }-a y^{3}+c y^{2}+3 y b \,x^{2}=-2 b \,x^{3}} \]
program solution
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {\left (-{\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{\frac {2}{3}}+\left (c x -c_{1} \right ) a 12^{\frac {1}{3}}\right ) 12^{\frac {1}{3}}}{6 {\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{\frac {1}{3}} a} \\ y \left (x \right ) &= -\frac {3^{\frac {1}{3}} 2^{\frac {2}{3}} \left (\left (1+i \sqrt {3}\right ) {\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{\frac {2}{3}}+a \left (i 3^{\frac {5}{6}}-3^{\frac {1}{3}}\right ) \left (c x -c_{1} \right ) 2^{\frac {2}{3}}\right )}{12 {\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{\frac {1}{3}} a} \\ y \left (x \right ) &= \frac {3^{\frac {1}{3}} \left (\left (i \sqrt {3}-1\right ) {\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{\frac {2}{3}}+\left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) a \left (c x -c_{1} \right ) 2^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{12 {\left (\left (-9 b \,x^{3}+9 c_{1} x +\sqrt {\frac {81 a \,b^{2} x^{6}-162 a b c_{1} x^{4}+12 c^{3} x^{3}+81 a \,c_{1}^{2} x^{2}-36 c^{2} c_{1} x^{2}+36 c \,c_{1}^{2} x -12 c_{1}^{3}}{a}}\right ) a^{2}\right )}^{\frac {1}{3}} a} \\ \end{align*}

Problem 8650

ODE	\[ \boxed {x y^{3} y^{\prime }+y^{4}=x \sin \left (x \right )} \]
program solution	\[ \left (x^{4}-12 x^{2}+24\right ) \cos \left (x \right )+4 \left (-x^{3}+6 x \right ) \sin \left (x \right )+\frac {y^{4} x^{4}}{4} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {{\left (4 \left (-x^{4}+12 x^{2}-24\right ) \cos \left (x \right )+16 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_{1} \right )}^{\frac {1}{4}}}{x} \\ y \left (x \right ) &= -\frac {{\left (4 \left (-x^{4}+12 x^{2}-24\right ) \cos \left (x \right )+16 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_{1} \right )}^{\frac {1}{4}}}{x} \\ y \left (x \right ) &= -\frac {i {\left (4 \left (-x^{4}+12 x^{2}-24\right ) \cos \left (x \right )+16 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_{1} \right )}^{\frac {1}{4}}}{x} \\ y \left (x \right ) &= \frac {i {\left (4 \left (-x^{4}+12 x^{2}-24\right ) \cos \left (x \right )+16 \left (x^{3}-6 x \right ) \sin \left (x \right )+c_{1} \right )}^{\frac {1}{4}}}{x} \\ \end{align*}

Problem 8651

ODE	\[ \boxed {\left (2 y^{3} x -x^{4}\right ) y^{\prime }-y^{4}+2 x^{3} y=0} \]
program solution	\[ \frac {y^{3}+x^{3}}{x y} = c_{1} \] 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}} \left (\left (-i \sqrt {3}-1\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 ) x 2^{\frac {2}{3}} c_{1} \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}} \\ 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}}+x 2^{\frac {2}{3}} \left (i 3^{\frac {5}{6}}+3^{\frac {1}{3}}\right ) c_{1} \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}} \\ \end{align*}

Problem 8652

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

Problem 8653

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

Problem 8654

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

Problem 8655

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

Problem 8656

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

Problem 8657

ODE	\[ \boxed {\left (2 y^{3} x^{2}+y^{2} x^{2}-2 x \right ) y^{\prime }-2 y=1} \]
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} x \,{\mathrm e}^{\textit {\_Z}}+3 x \,{\mathrm e}^{\textit {\_Z}}+16\right )}}{2}-\frac {1}{2} \\ \end{align*}

Problem 8658

ODE	\[ \boxed {\left (10 y^{3} x^{2}-3 y^{2}-2\right ) y^{\prime }+5 y^{4} x=-x} \]
program solution	\[ \frac {x^{2} \left (5 y^{4}+1\right )}{2}-y^{3}-2 y = c_{1} \] 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 8659

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

Problem 8660

ODE	\[ \boxed {\left (2 y^{3} x^{3}-x \right ) y^{\prime }+2 y^{3} x^{3}-y=0} \]
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 8661

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

Problem 8662

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

Problem 8663

ODE	\[ \boxed {\left (y^{4} x +2 y^{3} x^{2}+2 y+x \right ) y^{\prime }+y^{5}+y=0} \]
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}}\, c_{1} x +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}}\, c_{1} x +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}}\, c_{1} x +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}}\, c_{1} x +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}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}} x c_{1}} \\ y \left (x \right ) &= \frac {12 i \sqrt {3}\, c_{1} x^{2}+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}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {2}{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}}\, c_{1} x +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}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}-4}{12 x c_{1} \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}}\, c_{1} x +36 c_{1} x^{2}-8\right )^{\frac {1}{3}}} \\ \end{align*}

Problem 8664

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

Problem 8665

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

Problem 8666

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

Problem 8667

ODE	\[ \boxed {\frac {y^{\prime } f_{\nu }\left (x \right ) \left (-y+y^{p +1}\right )}{y-1}-\frac {g_{\nu }\left (x \right ) \left (-y+y^{q +1}\right )}{y-1}=0} \]
program solution	\[ \int _{}^{x}-\frac {g_{\nu }\left (\textit {\_a} \right )}{f_{\nu }\left (\textit {\_a} \right )}d \textit {\_a} +\frac {y \left (-y^{p} \operatorname {LerchPhi}\left (y^{q}, 1, \frac {p +1}{q}\right )+\operatorname {LerchPhi}\left (y^{q}, 1, \frac {1}{q}\right )\right )}{q} = c_{1} \] Verified OK. {(I*y^q)::positive}
Maple solution	\[ \frac {y \left (x \right )^{p +1} \operatorname {LerchPhi}\left (-y \left (x \right )^{q} \left (-1\right )^{\operatorname {csgn}\left (i y \left (x \right )^{q}\right )}, 1, \frac {p +1}{q}\right )-y \left (x \right ) \operatorname {LerchPhi}\left (-y \left (x \right )^{q} \left (-1\right )^{\operatorname {csgn}\left (i y \left (x \right )^{q}\right )}, 1, \frac {1}{q}\right )+q \left (\int \frac {g_{\nu }\left (x \right )}{f_{\nu }\left (x \right )}d x +c_{1} \right )}{q} = 0 \]

Problem 8668

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

Problem 8669

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

Problem 8670

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

Problem 8671

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

Problem 8672

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

Problem 8673

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

Problem 8674

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

Problem 8675

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

Problem 8676

ODE	\[ \boxed {\left (\frac {\operatorname {e1} \left (a +x \right )}{\left (\left (a +x \right )^{2}+y^{2}\right )^{\frac {3}{2}}}+\frac {\operatorname {e2} \left (x -a \right )}{\left (\left (x -a \right )^{2}+y^{2}\right )^{\frac {3}{2}}}\right ) y^{\prime }-y \left (\frac {\operatorname {e1}}{\left (\left (a +x \right )^{2}+y^{2}\right )^{\frac {3}{2}}}+\frac {\operatorname {e2}}{\left (\left (x -a \right )^{2}+y^{2}\right )^{\frac {3}{2}}}\right )=0} \]
program solution	\[ -\frac {\operatorname {e1} \left (a +x \right )}{\sqrt {y^{2}+a^{2}+2 a x +x^{2}}}+\frac {\operatorname {e2} \left (-x +a \right )}{\sqrt {y^{2}+a^{2}-2 a x +x^{2}}} = c_{1} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 8677

ODE	\[ \boxed {\left (x \,{\mathrm e}^{y}+{\mathrm e}^{x}\right ) y^{\prime }+{\mathrm e}^{y}+{\mathrm e}^{x} y=0} \]
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 8678

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

Problem 8679

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

Problem 8680

ODE	\[ \boxed {\left (\ln \left (y\right )+2 x -1\right ) y^{\prime }-2 y=0} \]
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 8681

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

Problem 8682

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

Problem 8683

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

Problem 8684

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

Problem 8685

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

Problem 8686

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

Problem 8687

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

Problem 8688

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

Problem 8689

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

Problem 8690

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

Problem 8691

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

Problem 8692

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

Problem 8693

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

Problem 8694

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

Problem 8695

ODE	\[ \boxed {3 y^{\prime } \sin \left (y\right ) \sin \left (x \right )+5 \cos \left (x \right )^{4} y=0} \]
program solution	\[ -\frac {\cos \left (x \right )^{3}}{3}-\cos \left (x \right )-\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )-\frac {3 \,\operatorname {Si}\left (y\right )}{5} = c_{1} \] Verified OK.
Maple solution	\[ \frac {3 \,\operatorname {Si}\left (y \left (x \right )\right )}{5}+c_{1} +\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+\frac {\cos \left (x \right )^{3}}{3}+\cos \left (x \right ) = 0 \]

Problem 8696

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

Problem 8697

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

Problem 8698

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

Problem 8699

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

Problem 8700

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