2.17.86 Problems 8501 to 8600

Problem 8501

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

Problem 8502

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

Problem 8503

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

Problem 8504

ODE	\[ \boxed {3 \left (x^{2}-4\right ) y^{\prime }+y^{2}-y x=3} \]
program solution	\[ y = \frac {-2 x \operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, \frac {x}{2}\right )-2 \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {x}{2}\right ) c_{3} x +3 \operatorname {LegendreQ}\left (\frac {5}{6}, \frac {1}{3}, \frac {x}{2}\right )+3 \operatorname {LegendreP}\left (\frac {5}{6}, \frac {1}{3}, \frac {x}{2}\right ) c_{3}}{\operatorname {LegendreQ}\left (-\frac {1}{6}, \frac {1}{3}, \frac {x}{2}\right )+\operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {x}{2}\right ) c_{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {\left (\left (x -2\right ) \left (-2 x -4\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{6}, \frac {1}{6}\right ], \left [-\frac {1}{3}\right ], -\frac {4}{x -2}\right )+24 \operatorname {hypergeom}\left (\left [\frac {5}{6}, \frac {7}{6}\right ], \left [\frac {7}{3}\right ], -\frac {4}{x -2}\right ) c_{1} \right ) \left (x +2\right )^{2}}{\left (-2 x -4\right )^{\frac {1}{3}} \left (x -2\right ) \left (x +2\right )^{2} \operatorname {hypergeom}\left (\left [-\frac {1}{6}, \frac {1}{6}\right ], \left [-\frac {1}{3}\right ], -\frac {4}{x -2}\right )+32 \left (x -\frac {5}{4}\right ) c_{1} \left (x +2\right )^{2} \operatorname {hypergeom}\left (\left [\frac {5}{6}, \frac {7}{6}\right ], \left [\frac {7}{3}\right ], -\frac {4}{x -2}\right )+4 \left (x -2\right )^{2} \left (\frac {x +2}{x -2}\right )^{\frac {1}{6}} \left (\left (-2 x -4\right )^{\frac {1}{3}} \left (x +2\right ) \operatorname {HeunCPrime}\left (0, -\frac {4}{3}, -\frac {1}{3}, 0, \frac {25}{36}, \frac {4}{x +2}\right )+24 \operatorname {HeunCPrime}\left (0, \frac {4}{3}, -\frac {1}{3}, 0, \frac {25}{36}, \frac {4}{x +2}\right ) c_{1} \right )} \]

Problem 8505

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

Problem 8506

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

Problem 8507

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

Problem 8508

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

Problem 8509

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

Problem 8510

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

Problem 8511

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

Problem 8512

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

Problem 8513

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

Problem 8514

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

Problem 8515

ODE	\[ \boxed {3 x \left (x^{2}-1\right ) y^{\prime }+x y^{2}-\left (x^{2}+1\right ) y=3 x} \]
program solution	\[ y = \frac {6 \pi \sqrt {3}\, \left (x^{\frac {2}{3}}+x^{\frac {8}{3}}\right ) \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )+27 c_{3} \left (x^{2}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right )^{2} \left (x^{2}+1\right ) \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )-14 \pi \sqrt {3}\, \left (x^{\frac {2}{3}}-x^{\frac {8}{3}}\right ) \operatorname {LegendreP}\left (\frac {5}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )+27 \operatorname {LegendreP}\left (\frac {5}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) \Gamma \left (\frac {2}{3}\right )^{2} c_{3} \left (x^{2}\right )^{\frac {1}{3}} \left (x -1\right ) \left (x +1\right )}{18 x \left (c_{3} \left (x^{2}\right )^{\frac {1}{3}} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) \Gamma \left (\frac {2}{3}\right )^{2}+\frac {2 x^{\frac {2}{3}} \pi \sqrt {3}\, \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )}{9}\right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {80 c_{1} \sqrt {3}\, \pi \left (x^{2}-\frac {2}{5}\right ) \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )+315 \Gamma \left (\frac {2}{3}\right ) \left (\frac {24 \left (x^{2}\right )^{\frac {1}{3}} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) \Gamma \left (\frac {2}{3}\right ) x^{\frac {4}{3}}}{35}+\left (x^{2}\right )^{\frac {1}{6}} \left (-x^{2}+1\right )^{\frac {5}{6}} \left (\left (x^{4}-x^{2}\right ) c_{1} \operatorname {hypergeom}\left (\left [\frac {11}{6}, \frac {13}{6}\right ], \left [\frac {7}{3}\right ], x^{2}\right )-\frac {6 \operatorname {hypergeom}\left (\left [\frac {3}{2}, \frac {11}{6}\right ], \left [\frac {5}{3}\right ], x^{2}\right ) \left (x^{\frac {4}{3}}-x^{\frac {10}{3}}\right )}{7}\right )\right )}{x^{\frac {1}{3}} \left (16 x^{\frac {2}{3}} \pi \sqrt {3}\, \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) c_{1} +72 \left (x^{2}\right )^{\frac {1}{3}} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) \Gamma \left (\frac {2}{3}\right )^{2}\right )} \]

Problem 8516

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

Problem 8517

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

Problem 8518

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

Problem 8519

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

Problem 8520

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

Problem 8521

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

Problem 8522

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

Problem 8523

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

Problem 8524

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

Problem 8525

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

Problem 8526

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

Problem 8527

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

Problem 8528

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

Problem 8529

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

Problem 8530

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

Problem 8531

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

Problem 8532

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

Problem 8533

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

Problem 8534

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

Problem 8535

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

Problem 8536

ODE	\[ \boxed {\left (a \sin \left (x \right )^{2}+b \right ) y^{\prime }+a y \sin \left (2 x \right )=-A x \left (a \sin \left (x \right )^{2}+c \right )} \]
program solution	\[ y = -\frac {2 A \sin \left (2 x \right ) a x -2 A a \,x^{2}-4 A c \,x^{2}+A \cos \left (2 x \right ) a -A a +8 c_{1}}{4 \left (a \cos \left (2 x \right )-a -2 b \right )} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-\cos \left (2 x \right ) A a -2 A x a \sin \left (2 x \right )+2 x^{2} \left (a +2 c \right ) A -8 c_{1}}{4 a \cos \left (2 x \right )-4 a -8 b} \]

Problem 8537

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

Problem 8538

ODE	\[ \boxed {f \left (x \right ) y^{\prime }+g \left (x \right ) s \left (y\right )=-h \left (x \right )} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 8539

ODE	\[ \boxed {y^{\prime } y+y=-x^{3}} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 8540

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

Problem 8541

ODE	\[ \boxed {y^{\prime } y+a y=-\frac {\left (a^{2}-1\right ) x}{4}-b \,x^{n}} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 8542

ODE	\[ \boxed {y^{\prime } y+a y=-b \,{\mathrm e}^{x}+2 a} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 8543

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

Problem 8544

ODE	\[ \boxed {y^{\prime } y+a y^{2}=b \cos \left (x +c \right )} \]
program solution	\[ \frac {{\mathrm e}^{2 a x} \left (4 y^{2} a^{2}-4 b \cos \left (x +c \right ) a -2 b \sin \left (x +c \right )+y^{2}\right )}{8 a^{2}+2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= \frac {\sqrt {16 c_{1} \left (a^{2}+\frac {1}{4}\right )^{2} {\mathrm e}^{-2 a x}+16 \left (a \cos \left (x +c \right )+\frac {\sin \left (x +c \right )}{2}\right ) \left (a^{2}+\frac {1}{4}\right ) b}}{4 a^{2}+1} \\ y \left (x \right ) &= -\frac {\sqrt {16 c_{1} \left (a^{2}+\frac {1}{4}\right )^{2} {\mathrm e}^{-2 a x}+16 \left (a \cos \left (x +c \right )+\frac {\sin \left (x +c \right )}{2}\right ) \left (a^{2}+\frac {1}{4}\right ) b}}{4 a^{2}+1} \\ \end{align*}

Problem 8545

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

Problem 8546

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

Problem 8547

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

Problem 8548

ODE	\[ \boxed {y^{\prime } y+f \left (y^{2}+x^{2}\right ) g \left (x \right )=-x} \]
program solution
Maple solution	\[ \int _{\textit {\_b}}^{y \left (x \right )}\frac {\textit {\_a}}{f \left (\textit {\_a}^{2}+x^{2}\right )}d \textit {\_a} +\int g \left (x \right )d x -c_{1} = 0 \]

Problem 8549

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

Problem 8550

ODE	\[ \boxed {\left (y+x -1\right ) y^{\prime }-y=-2 x -3} \]
program solution	\[ \frac {\ln \left (3 y^{2}+6 x^{2}-10 y+8 x +11\right )}{3}+\frac {\sqrt {2}\, \arctan \left (\frac {\left (3 y-5\right ) \sqrt {2}}{6 x +4}\right )}{3} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {5}{3}+\frac {\tan \left (\operatorname {RootOf}\left (\sqrt {2}\, \ln \left (2\right )+\sqrt {2}\, \ln \left (\sec \left (\textit {\_Z} \right )^{2} \left (3 x +2\right )^{2}\right )+2 c_{1} \sqrt {2}-2 \textit {\_Z} \right )\right ) \sqrt {2}\, \left (-3 x -2\right )}{3} \]

Problem 8551

ODE	\[ \boxed {\left (y+2 x -2\right ) y^{\prime }-y=-x -1} \]
program solution	\[ \frac {\ln \left (3 x^{2}+\left (3 y-6\right ) x +3 y^{2}-9 y+7\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (2 y-3+x \right ) \sqrt {3}}{3 x -1}\right ) = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3}{2}-\frac {x}{2}+\frac {\sqrt {3}\, \left (3 x -1\right ) \tan \left (\operatorname {RootOf}\left (\sqrt {3}\, \ln \left (3\right )-2 \sqrt {3}\, \ln \left (2\right )+\sqrt {3}\, \ln \left (\sec \left (\textit {\_Z} \right )^{2} \left (3 x -1\right )^{2}\right )+2 \sqrt {3}\, c_{1} +6 \textit {\_Z} \right )\right )}{6} \]

Problem 8552

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

Problem 8553

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

Problem 8554

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

Problem 8555

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

Problem 8556

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

Problem 8557

ODE	\[ \boxed {\left (2 y+x +1\right ) y^{\prime }-2 y=x -1} \]
program solution	\[ y = -\frac {x}{2}+\frac {2 \operatorname {LambertW}\left (\frac {{\mathrm e}^{\frac {9 x}{4}-\frac {1}{4}+\frac {9 c_{1}}{2}}}{4}\right )}{3}+\frac {1}{6} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x}{2}+\frac {2 \operatorname {LambertW}\left (\frac {c_{1} {\mathrm e}^{\frac {9 x}{4}-\frac {1}{4}}}{4}\right )}{3}+\frac {1}{6} \]

Problem 8558

ODE	\[ \boxed {\left (2 y+x +7\right ) y^{\prime }-y=-4-2 x} \]
program solution	\[ \frac {\ln \left (y^{2}+x^{2}+4 y+6 x +13\right )}{2}+\frac {\arctan \left (\frac {y+2}{x +3}\right )}{2} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -2+\tan \left (\operatorname {RootOf}\left (\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )-\textit {\_Z} +2 \ln \left (x +3\right )+2 c_{1} \right )\right ) \left (-x -3\right ) \]

Problem 8559

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

Problem 8560

ODE	\[ \boxed {\left (2 y-6 x \right ) y^{\prime }-y=-3 x -2} \]
program solution	\[ y = \frac {{\mathrm e}^{-\operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {25 x}{4}-1-\frac {25 c_{1}}{4}}}{2}\right )+\frac {25 x}{4}-1-\frac {25 c_{1}}{4}}}{5}+3 x -\frac {2}{5} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {2 \operatorname {LambertW}\left (-\frac {{\mathrm e}^{\frac {25 x}{4}-1-\frac {25 c_{1}}{4}}}{2}\right )}{5}+3 x -\frac {2}{5} \]

Problem 8561

ODE	\[ \boxed {\left (4 y+2 x +3\right ) y^{\prime }-2 y=x +1} \]
program solution	\[ y = \frac {\operatorname {LambertW}\left ({\mathrm e}^{8 x +5+16 c_{1}}\right )}{8}-\frac {x}{2}-\frac {5}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = -\frac {x}{2}+\frac {\operatorname {LambertW}\left (c_{1} {\mathrm e}^{5+8 x}\right )}{8}-\frac {5}{8} \]

Problem 8562

ODE	\[ \boxed {\left (4 y-2 x -3\right ) y^{\prime }+2 y=x +1} \]
program solution	\[ y = \frac {x}{2}-\frac {\operatorname {LambertW}\left (-{\mathrm e}^{8 x +5-16 c_{1}}\right )}{8}+\frac {5}{8} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x}{2}-\frac {\operatorname {LambertW}\left (-c_{1} {\mathrm e}^{5+8 x}\right )}{8}+\frac {5}{8} \]

Problem 8563

ODE	\[ \boxed {\left (4 y-3 x -5\right ) y^{\prime }-3 y=-2-7 x} \]
program solution	\[ \frac {x \left (7 x -6 y+4\right )}{2}+2 y^{2}-5 y = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-\sqrt {4-6859 \left (x -\frac {7}{19}\right )^{2} c_{1}^{2}}+\left (57 x +95\right ) c_{1}}{76 c_{1}} \]

Problem 8564

ODE	\[ \boxed {\left (4 y+11 x -11\right ) y^{\prime }-25 y=8 x -62} \]
program solution	\[ -\frac {2 \ln \left (2 y-5+x \right )}{279}+\frac {2 \ln \left (y-2-4 x \right )}{93} = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {4 \left (x +\frac {1}{2}\right ) \left (i \sqrt {3}-1\right ) {\left (708588 \sqrt {\left (-\frac {32}{177147}+\left (x -\frac {1}{9}\right )^{2} c_{1} \right ) \left (x -\frac {1}{9}\right )^{2} c_{1}}[phone] \left (x -\frac {1}{9}\right )^{2} c_{1} \right )}^{\frac {2}{3}}+\left (-76 x +28\right ) {\left (708588 \sqrt {\left (-\frac {32}{177147}+\left (x -\frac {1}{9}\right )^{2} c_{1} \right ) \left (x -\frac {1}{9}\right )^{2} c_{1}}[phone] \left (x -\frac {1}{9}\right )^{2} c_{1} \right )}^{\frac {1}{3}}-64 \left (x +\frac {1}{2}\right ) \left (1+i \sqrt {3}\right )}{i \sqrt {3}\, {\left (708588 \sqrt {\left (-\frac {32}{177147}+\left (x -\frac {1}{9}\right )^{2} c_{1} \right ) \left (x -\frac {1}{9}\right )^{2} c_{1}}[phone] \left (x -\frac {1}{9}\right )^{2} c_{1} \right )}^{\frac {2}{3}}-16 i \sqrt {3}-{\left (708588 \sqrt {\left (-\frac {32}{177147}+\left (x -\frac {1}{9}\right )^{2} c_{1} \right ) \left (x -\frac {1}{9}\right )^{2} c_{1}}[phone] \left (x -\frac {1}{9}\right )^{2} c_{1} \right )}^{\frac {2}{3}}+8 {\left (708588 \sqrt {\left (-\frac {32}{177147}+\left (x -\frac {1}{9}\right )^{2} c_{1} \right ) \left (x -\frac {1}{9}\right )^{2} c_{1}}[phone] \left (x -\frac {1}{9}\right )^{2} c_{1} \right )}^{\frac {1}{3}}-16} \]

Problem 8565

ODE	\[ \boxed {\left (12 y-5 x -8\right ) y^{\prime }-5 y=-2 x -3} \]
program solution	\[ x \left (x -5 y+3\right )+6 y^{2}-8 y = c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-\sqrt {\left (x +4\right )^{2} c_{1}^{2}+24}+\left (5 x +8\right ) c_{1}}{12 c_{1}} \]

Problem 8566

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

Problem 8567

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

Problem 8568

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

Problem 8569

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

Problem 8570

ODE	\[ \boxed {y x y^{\prime }-y^{2}+y x=-x^{3}+2 x^{2}} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 8571

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

Problem 8572

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

Problem 8573

ODE	\[ \boxed {x \left (y+a \right ) y^{\prime }+y b=-c x} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 8574

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

Problem 8575

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

Problem 8576

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

Problem 8577

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

Problem 8578

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

Problem 8579

ODE	\[ \boxed {x \left (2 y+x -1\right ) y^{\prime }-y \left (y+2 x +1\right )=0} \]
program solution	\[ -\frac {3 \left (x -y-1\right )}{\left (y x \right )^{\frac {1}{3}}} = c_{1} \] 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}}}-1+x \\ y \left (x \right ) &= \frac {\frac {3 \,5^{\frac {1}{3}} \left (-i \sqrt {3}-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 {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}+\left (i \sqrt {3}-1\right ) x 5^{\frac {2}{3}}\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}+x 5^{\frac {2}{3}} \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 8580

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

Problem 8581

ODE	\[ \boxed {\left (2 y x +4 x^{3}\right ) y^{\prime }+y^{2}+112 y x^{2}=0} \]
program solution	\[ \frac {\ln \left (y\right )}{30}+\frac {11 \ln \left (24 x^{2}+y\right )}{30} = -\frac {\ln \left (x \right )}{5}+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{x^{28} \operatorname {RootOf}\left (x^{30} \textit {\_Z}^{360}-24 x^{30} \textit {\_Z}^{330}-c_{1} \right )^{330}} \]

Problem 8582

ODE	\[ \boxed {x \left (3 y+2 x \right ) y^{\prime }+3 \left (y+x \right )^{2}=0} \]
program solution	\[ \frac {3 x^{4}}{4}+2 x^{3} y+\frac {3 y^{2} x^{2}}{2} = c_{1} \] 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 8583

ODE	\[ \boxed {\left (3 x +2\right ) \left (y-2 x -1\right ) y^{\prime }-y^{2}+y x=7 x^{2}+9 x +3} \]
program solution	\[ \frac {\ln \left (2 y-4-7 x \right )}{6}+\frac {\ln \left (x +y+1\right )}{3} = \frac {\ln \left (3 x +2\right )}{6}+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\frac {14 \left (x +\frac {2}{3}\right ) \left (-\frac {4}{1701}+\left (x^{2}+\frac {26}{21} x +\frac {8}{21}\right ) c_{1}^{2}\right ) {\left (2 \sqrt {-2187 \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) \left (x +\frac {2}{3}\right )^{2} c_{1}^{2}}+\left (-729 x^{3}-1458 x^{2}-972 x -216\right ) c_{1}^{3}+\left (6 x +4\right ) c_{1} \right )}^{\frac {2}{3}}}{27}+21 \left (\frac {\left (-\frac {2 \sqrt {-2187 \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) \left (x +\frac {2}{3}\right )^{2} c_{1}^{2}}}{2187}+\left (-\frac {2}{729}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) \left (x +\frac {2}{3}\right ) c_{1} \right ) \left (1+i \sqrt {3}\right ) {\left (2 \sqrt {-2187 \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) \left (x +\frac {2}{3}\right )^{2} c_{1}^{2}}+\left (-729 x^{3}-1458 x^{2}-972 x -216\right ) c_{1}^{3}+\left (6 x +4\right ) c_{1} \right )}^{\frac {1}{3}}}{9}+\left (i \sqrt {3}-1\right ) \left (x +\frac {2}{3}\right ) \left (-\frac {4 \sqrt {-2187 \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) \left (x +\frac {2}{3}\right )^{2} c_{1}^{2}}}{2187}+\left (-\frac {2}{27}+\left (x +\frac {2}{3}\right ) c_{1} \right ) \left (x +\frac {2}{3}\right ) \left (\frac {2}{27}+\left (x +\frac {2}{3}\right ) c_{1} \right ) c_{1} \right ) c_{1} \right ) \left (x +\frac {4}{7}\right )}{{\left (\frac {\left (1-i \sqrt {3}\right ) {\left (2 \sqrt {-2187 \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) \left (x +\frac {2}{3}\right )^{2} c_{1}^{2}}+\left (-729 x^{3}-1458 x^{2}-972 x -216\right ) c_{1}^{3}+\left (6 x +4\right ) c_{1} \right )}^{\frac {2}{3}}}{81}+\left (x +\frac {2}{3}\right ) \left (\frac {2 {\left (2 \sqrt {-2187 \left (-\frac {1}{243}+\left (x +\frac {2}{3}\right )^{2} c_{1}^{2}\right ) \left (x +\frac {2}{3}\right )^{2} c_{1}^{2}}+\left (-729 x^{3}-1458 x^{2}-972 x -216\right ) c_{1}^{3}+\left (6 x +4\right ) c_{1} \right )}^{\frac {1}{3}}}{9}+\left (x +\frac {2}{3}\right ) c_{1} \left (1+i \sqrt {3}\right )\right ) c_{1} \right )}^{2}} \]

Problem 8584

ODE	\[ \boxed {\left (6 y x +x^{2}+3\right ) y^{\prime }+3 y^{2}+2 y x=-2 x} \]
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 8585

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

Problem 8586

ODE	\[ \boxed {\left (B x y+A \,x^{2}+a x +y b +c \right ) y^{\prime }+A x y+\beta y=B g \left (x \right )^{2}-x \alpha -\gamma } \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 8587

ODE	\[ \boxed {\left (y x^{2}-1\right ) y^{\prime }+x y^{2}=1} \]
program solution	\[ \frac {y^{2} x^{2}}{2}-x -y = c_{1} \] 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 8588

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

Problem 8589

ODE	\[ \boxed {\left (y x^{2}-1\right ) y^{\prime }+8 x y^{2}=8} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 8590

ODE	\[ \boxed {x \left (y x -2\right ) y^{\prime }+y^{3} x^{2}+x y^{2}-2 y=0} \]
program solution	\[ \ln \left (x \right )-\frac {1}{x y}+\frac {1}{y^{2} x^{2}} = c_{1} \] 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 8591

ODE	\[ \boxed {x \left (y x -3\right ) y^{\prime }+x y^{2}-y=0} \]
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 8592

ODE	\[ \boxed {x^{2} \left (y-1\right ) y^{\prime }+\left (x -1\right ) y=0} \]
program solution	\[ y = -\operatorname {LambertW}\left (-{\mathrm e}^{\frac {\ln \left (x \right ) x +x c_{1} +1}{x}}\right ) \] 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 8593

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

Problem 8594

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

Problem 8595

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

Problem 8596

ODE	\[ \boxed {\left (2 y x^{2}+x \right ) y^{\prime }-y^{3} x^{2}+2 x y^{2}+y=0} \]
program solution	\[ -\ln \left (x \right )-\frac {2}{x y}-\frac {1}{2 y^{2} x^{2}} = c_{1} \] 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 8597

ODE	\[ \boxed {\left (2 y x^{2}-x \right ) y^{\prime }-2 x y^{2}-y=0} \]
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 8598

ODE	\[ \boxed {\left (2 y x^{2}-x^{3}\right ) y^{\prime }+y^{3}-4 x y^{2}=-2 x^{3}} \]
program solution	\[ -\frac {\ln \left (y+x \right )}{2}+\ln \left (y-2 x \right )-\frac {\ln \left (y-x \right )}{2} = -\ln \left (x \right )+c_{1} \] 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 8599

ODE	\[ \boxed {y^{\prime } y+3 y^{2} x^{2}=-2 x^{3}-7} \]
program solution	\[ -\frac {10 \left (-\frac {3 \Gamma \left (\frac {2}{3}\right ) \left (\left (\frac {3 y^{2}}{2}+x \right ) {\mathrm e}^{2 x^{3}}-\frac {3 y^{2}}{2}\right ) \left (-x^{3}\right )^{\frac {1}{3}}}{10}+2^{\frac {2}{3}} x \left (\Gamma \left (\frac {1}{3}, -2 x^{3}\right ) \Gamma \left (\frac {2}{3}\right )-\frac {2 \pi \sqrt {3}}{3}\right )\right )}{9 \left (-x^{3}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}\right )}+\frac {y^{2}}{2} = c_{1} \] Verified OK.
Maple solution	\begin{align*} y \left (x \right ) &= -\frac {2^{\frac {5}{6}} \sqrt {3}\, \sqrt {-80 \left (\frac {9 \Gamma \left (\frac {2}{3}\right ) \left (-\frac {3 \,{\mathrm e}^{-2 x^{3}} c_{1}}{2}+x \right ) 2^{\frac {1}{3}} \left (-x^{3}\right )^{\frac {1}{3}}}{40}+x \,{\mathrm e}^{-2 x^{3}} \left (\pi \sqrt {3}-\frac {3 \Gamma \left (\frac {1}{3}, -2 x^{3}\right ) \Gamma \left (\frac {2}{3}\right )}{2}\right )\right ) \left (-x^{3}\right )^{\frac {1}{3}}}}{18 \left (-x^{3}\right )^{\frac {1}{3}} \sqrt {\Gamma \left (\frac {2}{3}\right )}} \\ y \left (x \right ) &= \frac {2^{\frac {5}{6}} \sqrt {3}\, \sqrt {-80 \left (\frac {9 \Gamma \left (\frac {2}{3}\right ) \left (-\frac {3 \,{\mathrm e}^{-2 x^{3}} c_{1}}{2}+x \right ) 2^{\frac {1}{3}} \left (-x^{3}\right )^{\frac {1}{3}}}{40}+x \,{\mathrm e}^{-2 x^{3}} \left (\pi \sqrt {3}-\frac {3 \Gamma \left (\frac {1}{3}, -2 x^{3}\right ) \Gamma \left (\frac {2}{3}\right )}{2}\right )\right ) \left (-x^{3}\right )^{\frac {1}{3}}}}{18 \left (-x^{3}\right )^{\frac {1}{3}} \sqrt {\Gamma \left (\frac {2}{3}\right )}} \\ \end{align*}

Problem 8600

ODE	\[ \boxed {2 x \left (x^{3} y+1\right ) y^{\prime }+\left (3 x^{3} y-1\right ) y=0} \]
program solution	\[ -\frac {2 \ln \left (y\right )}{7}-\frac {8 \ln \left (3 x^{3} y+7\right )}{21} = -\frac {\ln \left (x \right )}{7}+c_{1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\operatorname {RootOf}\left (\textit {\_Z}^{98} c_{1} -14 \textit {\_Z}^{77} c_{1} +49 \textit {\_Z}^{56} c_{1} -9 x^{7}\right )^{21}-7}{3 x^{3}} \]