2.17.97 Problems 9601 to 9700

Problem 9601

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }-\left (-4 k x +4 m^{2}+x^{2}-1\right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (2 m , 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (2 m , 2 \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {WhittakerM}\left (k , m , x\right )+c_{2} \operatorname {WhittakerW}\left (k , m , x\right ) \]

Problem 9602

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

Problem 9603

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

Problem 9604

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

Problem 9605

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

Problem 9606

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+4 y^{\prime } x +f \left (x \right ) y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9607

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+5 y^{\prime } x -y=\ln \left (x \right )} \]
program solution	\[ y = c_{1} x^{-\frac {1}{8}-\frac {\sqrt {17}}{8}}+\frac {4 c_{2} \sqrt {17}\, x^{-\frac {1}{8}+\frac {\sqrt {17}}{8}}}{17}-\ln \left (x \right )-1 \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{-\frac {1}{8}+\frac {\sqrt {17}}{8}} c_{2} +x^{-\frac {1}{8}-\frac {\sqrt {17}}{8}} c_{1} -\ln \left (x \right )-1 \]

Problem 9608

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+8 y^{\prime } x -\left (4 x^{2}+12 x +3\right ) y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{x} \sqrt {x}-\frac {c_{2} {\mathrm e}^{-x} \left (4 \,\operatorname {expIntegral}_{1}\left (2 x \right ) x^{2} {\mathrm e}^{2 x}-2 x +1\right )}{2 x^{\frac {3}{2}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {-4 \,{\mathrm e}^{x} \operatorname {expIntegral}_{1}\left (2 x \right ) c_{2} x^{2}+\left (-1+2 x \right ) c_{2} {\mathrm e}^{-x}+c_{1} x^{2} {\mathrm e}^{x}}{x^{\frac {3}{2}}} \]

Problem 9609

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

Problem 9610

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

Problem 9611

ODE	\[ \boxed {4 x^{2} y^{\prime \prime }+4 x^{2} \ln \left (x \right ) y^{\prime }+\left (\ln \left (x \right )^{2} x^{2}+2 x -8\right ) y=4 x^{2} \sqrt {{\mathrm e}^{x} x^{-x}}} \]
program solution	\[ y = \frac {x^{-\frac {x}{2}-1} {\mathrm e}^{\frac {x}{2}} \left (c_{2} x^{3}+3 c_{1} \right )}{3}+\frac {x^{2} \sqrt {{\mathrm e}^{x} x^{-x}}\, \left (-1+3 \ln \left (x \right )\right )}{9} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{2} \left (\ln \left (x \right )-\frac {1}{3}\right ) \sqrt {x^{-x} {\mathrm e}^{x}}}{3}+{\mathrm e}^{\frac {x}{2}} \left (c_{1} x^{-\frac {x}{2}+2}+c_{2} x^{-\frac {x}{2}-1}\right ) \]

Problem 9612

ODE	\[ \boxed {\left (2 x +1\right )^{2} y^{\prime \prime }-2 \left (2 x +1\right ) y^{\prime }-12 y=3 x +1} \]
program solution	\[ y = \frac {3072 c_{2} x^{4}+6144 c_{2} x^{3}+\left (4608 c_{2} -72\right ) x^{2}+\left (1536 c_{2} -56\right ) x -24 c_{1} +192 c_{2} -7}{384 x +192} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{2 x +1}+\left (2 x +1\right )^{3} c_{2} +\frac {-72 x^{2}-56 x -7}{384 x +192} \]

Problem 9613

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

Problem 9614

ODE	\[ \boxed {\left (-1+3 x \right )^{2} y^{\prime \prime }+3 \left (-1+3 x \right ) y^{\prime }-9 y=\ln \left (-1+3 x \right )^{2}} \]
program solution	\[ y = \frac {\left (-6 x +2\right ) \ln \left (-1+3 x \right )^{2}+162 c_{2} x^{2}+\left (-108 c_{2} -12\right ) x -3 c_{1} +18 c_{2} +4}{-18+54 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1}}{3 x -1}+\left (3 x -1\right ) c_{2} -\frac {\ln \left (3 x -1\right )^{2}}{9}-\frac {2}{9} \]

Problem 9615

ODE	\[ \boxed {9 x \left (x -1\right ) y^{\prime \prime }+3 \left (2 x -1\right ) y^{\prime }-20 y=0} \]
program solution	\[ y = \frac {c_{1} \left (6 x -1\right ) x^{\frac {1}{6}} \left (x -1\right )^{\frac {5}{6}}}{6 \left (x \left (x -1\right )\right )^{\frac {1}{6}}}-\frac {9 c_{2} \left (6 x -5\right ) \left (x -1\right )^{\frac {1}{6}} x^{\frac {5}{6}}}{5 \left (x \left (x -1\right )\right )^{\frac {1}{6}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \left (6 x -5\right ) x^{\frac {2}{3}}+c_{2} \left (6 x -1\right ) \left (x -1\right )^{\frac {2}{3}} \]

Problem 9616

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

Problem 9617

ODE	\[ \boxed {16 x^{2} y^{\prime \prime }+32 y^{\prime } x -\left (4 x +5\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (\sqrt {x}-1\right ) {\mathrm e}^{\sqrt {x}}}{x^{\frac {5}{4}}}+\frac {c_{2} \left (-\sqrt {x}-1\right ) {\mathrm e}^{-\sqrt {x}}}{x^{\frac {5}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} \left (\sqrt {x}+1\right ) {\mathrm e}^{-\sqrt {x}}+c_{1} {\mathrm e}^{\sqrt {x}} \left (\sqrt {x}-1\right )}{x^{\frac {5}{4}}} \]

Problem 9618

ODE	\[ \boxed {\left (27 x^{2}+4\right ) y^{\prime \prime }+27 y^{\prime } x -3 y=0} \]
program solution	\[ y = \frac {c_{1} \sqrt {2}\, \left (12 \sqrt {3}\, x +4 \sqrt {27 x^{2}+4}\right )^{\frac {1}{3}}}{4}+2 c_{2} \sqrt {2}\, \left (12 \sqrt {3}\, x +4 \sqrt {27 x^{2}+4}\right )^{\frac {1}{3}} \left (\int \frac {1}{\sqrt {27 x^{2}+4}\, \left (12 \sqrt {3}\, x +4 \sqrt {27 x^{2}+4}\right )^{\frac {2}{3}}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} \sinh \left (\frac {\operatorname {arcsinh}\left (\frac {3 \sqrt {3}\, x}{2}\right )}{3}\right )+c_{2} \cosh \left (\frac {\operatorname {arcsinh}\left (\frac {3 \sqrt {3}\, x}{2}\right )}{3}\right ) \]

Problem 9619

ODE	\[ \boxed {48 x \left (x -1\right ) y^{\prime \prime }+\left (152 x -40\right ) y^{\prime }+53 y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\frac {13}{12}-\frac {\sqrt {10}}{12}, \frac {13}{12}+\frac {\sqrt {10}}{12}\right ], \left [\frac {5}{6}\right ], x\right )+c_{2} x^{\frac {1}{6}} \operatorname {hypergeom}\left (\left [\frac {5}{4}-\frac {\sqrt {10}}{12}, \frac {5}{4}+\frac {\sqrt {10}}{12}\right ], \left [\frac {7}{6}\right ], x\right ) \]

Problem 9620

ODE	\[ \boxed {50 x \left (x -1\right ) y^{\prime \prime }+25 \left (2 x -1\right ) y^{\prime }-2 y=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{\frac {5 i \pi \sqrt {-\operatorname {signum}\left (x -1\right )}-8 \sqrt {\operatorname {signum}\left (x -1\right )}\, \arcsin \left (\sqrt {x}\right )}{20 \sqrt {-\operatorname {signum}\left (x -1\right )}}} \left (-x \left (x -1\right )\right )^{\frac {1}{4}}}{\left (x \left (x -1\right )\right )^{\frac {1}{4}}}-\frac {i c_{2} {\mathrm e}^{\frac {5 i \pi \sqrt {-\operatorname {signum}\left (x -1\right )}-8 \sqrt {\operatorname {signum}\left (x -1\right )}\, \arcsin \left (\sqrt {x}\right )}{20 \sqrt {-\operatorname {signum}\left (x -1\right )}}} \left (-x \left (x -1\right )\right )^{\frac {1}{4}} \left (\int \frac {{\mathrm e}^{\frac {4 \sqrt {\operatorname {signum}\left (x -1\right )}\, \arcsin \left (\sqrt {x}\right )}{5 \sqrt {-\operatorname {signum}\left (x -1\right )}}}}{\sqrt {-x \left (x -1\right )}}d x \right )}{\left (x \left (x -1\right )\right )^{\frac {1}{4}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \left (\sqrt {x}+\sqrt {x -1}\right )^{\frac {4}{5}}+c_{2}}{\left (\sqrt {x}+\sqrt {x -1}\right )^{\frac {2}{5}}} \]

Problem 9621

ODE	\[ \boxed {144 x \left (x -1\right ) y^{\prime \prime }+\left (120 x -48\right ) y^{\prime }+y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \left (\operatorname {LegendreP}\left (-\frac {1}{2}, \frac {2}{3}, \sqrt {1-x}\right ) c_{1} +\operatorname {LegendreQ}\left (-\frac {1}{2}, \frac {2}{3}, \sqrt {1-x}\right ) c_{2} \right ) x^{\frac {1}{3}} \]

Problem 9622

ODE	\[ \boxed {144 x \left (x -1\right ) y^{\prime \prime }+\left (168 x -96\right ) y^{\prime }+y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \left (\operatorname {LegendreQ}\left (-\frac {1}{2}, \frac {1}{3}, \sqrt {1-x}\right ) c_{2} +\operatorname {LegendreP}\left (-\frac {1}{2}, \frac {1}{3}, \sqrt {1-x}\right ) c_{1} \right ) x^{\frac {1}{6}} \]

Problem 9623

ODE	\[ \boxed {a \,x^{2} y^{\prime \prime }+b x y^{\prime }+\left (c \,x^{2}+d x +f \right ) y=0} \]
program solution	\[ y = c_{1} x^{-\frac {b}{2 a}+\frac {1}{2}} \operatorname {BesselJ}\left (\frac {\sqrt {a^{2}-2 a b -4 f a +b^{2}}}{a}, 2 \sqrt {x}\right )+c_{2} x^{-\frac {b}{2 a}+\frac {1}{2}} \operatorname {BesselY}\left (\frac {\sqrt {a^{2}-2 a b -4 f a +b^{2}}}{a}, 2 \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{-\frac {b}{2 a}} \left (\operatorname {WhittakerM}\left (-\frac {i d}{2 \sqrt {a}\, \sqrt {c}}, \frac {\sqrt {a^{2}+\left (-2 b -4 f \right ) a +b^{2}}}{2 a}, \frac {2 i \sqrt {c}\, x}{\sqrt {a}}\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {i d}{2 \sqrt {a}\, \sqrt {c}}, \frac {\sqrt {a^{2}+\left (-2 b -4 f \right ) a +b^{2}}}{2 a}, \frac {2 i \sqrt {c}\, x}{\sqrt {a}}\right ) c_{2} \right ) \]

Problem 9624

ODE	\[ \boxed {\operatorname {a2} \,x^{2} y^{\prime \prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x \right ) y^{\prime }+\left (\operatorname {a0} \,x^{2}+\operatorname {b0} x +\operatorname {c0} \right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {\operatorname {a1} x}{2 \operatorname {a2}}} x^{-\frac {\operatorname {b1}}{2 \operatorname {a2}}} \left (\operatorname {WhittakerM}\left (-\frac {\operatorname {a1} \operatorname {b1} -2 \operatorname {a2} \operatorname {b0}}{2 \operatorname {a2} \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}}, \frac {\sqrt {\operatorname {a2}^{2}+\left (-2 \operatorname {b1} -4 \operatorname {c0} \right ) \operatorname {a2} +\operatorname {b1}^{2}}}{2 \operatorname {a2}}, \frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, x}{\operatorname {a2}}\right ) c_{1} +c_{2} \operatorname {WhittakerW}\left (-\frac {\operatorname {a1} \operatorname {b1} -2 \operatorname {a2} \operatorname {b0}}{2 \operatorname {a2} \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}}, \frac {\sqrt {\operatorname {a2}^{2}+\left (-2 \operatorname {b1} -4 \operatorname {c0} \right ) \operatorname {a2} +\operatorname {b1}^{2}}}{2 \operatorname {a2}}, \frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, x}{\operatorname {a2}}\right )\right ) \]

Problem 9625

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

Problem 9626

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

Problem 9627

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

Problem 9628

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

Problem 9629

ODE	\[ \boxed {\operatorname {A2} \left (x a +b \right )^{2} y^{\prime \prime }+\operatorname {A1} \left (x a +b \right ) y^{\prime }+\operatorname {A0} \left (x a +b \right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \left (a x +b \right )^{-\frac {-a \operatorname {A2} +\operatorname {A1}}{2 a \operatorname {A2}}} \left (\operatorname {BesselY}\left (\frac {a \operatorname {A2} -\operatorname {A1}}{a \operatorname {A2}}, 2 \sqrt {\operatorname {A0}}\, \sqrt {\frac {a x +b}{a^{2} \operatorname {A2}}}\right ) c_{2} +\operatorname {BesselJ}\left (\frac {a \operatorname {A2} -\operatorname {A1}}{a \operatorname {A2}}, 2 \sqrt {\operatorname {A0}}\, \sqrt {\frac {a x +b}{a^{2} \operatorname {A2}}}\right ) c_{1} \right ) \]

Problem 9630

ODE	\[ \boxed {\left (x^{2} a +b x +c \right ) y^{\prime \prime }+\left (d x +f \right ) y^{\prime }+g y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [-\frac {a -d +\sqrt {a^{2}+\left (-2 d -4 g \right ) a +d^{2}}}{2 a}, \frac {-a +d +\sqrt {a^{2}+\left (-2 d -4 g \right ) a +d^{2}}}{2 a}\right ], \left [\frac {d \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a -2 a f +b d}{2 a^{2} \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}}\right ], \frac {\left (-2 a^{2} x -a b \right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}+4 a c -b^{2}}{8 a c -2 b^{2}}\right )+c_{2} {\left (2 \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, x \,a^{2}+\sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, b a -4 a c +b^{2}\right )}^{\frac {a \left (a -\frac {d}{2}\right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}+a f -\frac {b d}{2}}{\sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a^{2}}} \operatorname {hypergeom}\left (\left [\frac {a \left (a -\sqrt {a^{2}+\left (-2 d -4 g \right ) a +d^{2}}\right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}+2 a f -b d}{2 \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a^{2}}, \frac {a \left (a +\sqrt {a^{2}+\left (-2 d -4 g \right ) a +d^{2}}\right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}+2 a f -b d}{2 \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a^{2}}\right ], \left [\frac {4 a^{2} \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}-d \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}\, a +2 a f -b d}{2 a^{2} \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}}\right ], \frac {\left (-2 a^{2} x -a b \right ) \sqrt {\frac {-4 a c +b^{2}}{a^{2}}}+4 a c -b^{2}}{8 a c -2 b^{2}}\right ) \]

Problem 9631

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

Problem 9632

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

Problem 9633

ODE	\[ \boxed {x^{3} y^{\prime \prime }+x^{2} y^{\prime }+\left (x^{2} a +b x +a \right ) y=0} \]
program solution	\[ y = c_{1} \operatorname {BesselJ}\left (2 \sqrt {-b}, 2 \sqrt {x}\right )+c_{2} \operatorname {BesselY}\left (2 \sqrt {-b}, 2 \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {HeunD}\left (0, 8 a +4 b , 0, 8 a -4 b , \frac {x +1}{x -1}\right ) \left (c_{1} +c_{2} \left (\int \frac {1}{x \operatorname {HeunD}\left (0, 8 a +4 b , 0, 8 a -4 b , \frac {x +1}{x -1}\right )^{2}}d x \right )\right ) \]

Problem 9634

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

Problem 9635

ODE	\[ \boxed {x^{3} y^{\prime \prime }-x^{2} y^{\prime }+x y=\ln \left (x \right )^{3}} \]
program solution	\[ y = x \left (c_{1} +c_{2} \ln \left (x \right )\right )+\frac {2 \ln \left (x \right )^{3}+6 \ln \left (x \right )^{2}+9 \ln \left (x \right )+6}{8 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {2 \ln \left (x \right )^{3}+6 \ln \left (x \right )^{2}+\left (8 c_{1} x^{2}+9\right ) \ln \left (x \right )+8 c_{2} x^{2}+6}{8 x} \]

Problem 9636

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

Problem 9637

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

Problem 9638

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

Problem 9639

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

Problem 9640

ODE	\[ \boxed {x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 \left (1+n \right ) x^{2}+2 n +1\right ) y^{\prime }-\left (v -n \right ) \left (v +n +1\right ) x y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = x^{-n} \left (\operatorname {LegendreQ}\left (v , n , \sqrt {x^{2}+1}\right ) c_{2} +\operatorname {LegendreP}\left (v , n , \sqrt {x^{2}+1}\right ) c_{1} \right ) \]

Problem 9641

ODE	\[ \boxed {x \left (x^{2}+1\right ) y^{\prime \prime }-\left (2 \left (-1+n \right ) x^{2}+2 n -1\right ) y^{\prime }+\left (v +n \right ) \left (n -1-v \right ) x y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = x^{n} \left (\operatorname {LegendreQ}\left (v , n , \sqrt {x^{2}+1}\right ) c_{2} +\operatorname {LegendreP}\left (v , n , \sqrt {x^{2}+1}\right ) c_{1} \right ) \]

Problem 9642

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

Problem 9643

ODE	\[ \boxed {x \left (x^{2}-1\right ) y^{\prime \prime }+\left (x^{2}-1\right ) y^{\prime }-x y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {EllipticE}\left (x \right )+c_{2} \left (\operatorname {EllipticCE}\left (x \right )-\operatorname {EllipticCK}\left (x \right )\right ) \]

Problem 9644

ODE	\[ \boxed {x \left (x^{2}-1\right ) y^{\prime \prime }+\left (3 x^{2}-1\right ) y^{\prime }+x y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {EllipticK}\left (x \right )+c_{2} \operatorname {EllipticCK}\left (x \right ) \]

Problem 9645

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

Problem 9646

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

Problem 9647

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

Problem 9648

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

Problem 9649

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

Problem 9650

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

Problem 9651

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

Problem 9652

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (\left (a +1+b \right ) x +\alpha +\beta -1\right ) y^{\prime }}{x \left (x -1\right )}+\frac {\left (a b x -\alpha \beta \right ) y}{x^{2} \left (x -1\right )}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \left (\operatorname {hypergeom}\left (\left [1-\alpha -b , 1-a -\alpha \right ], \left [1-\alpha +\beta \right ], x\right ) x^{\beta } c_{2} +\operatorname {hypergeom}\left (\left [1-b -\beta , 1-a -\beta \right ], \left [1+\alpha -\beta \right ], x\right ) x^{\alpha } c_{1} \right ) \left (x -1\right )^{1-a -\alpha -b -\beta } \]

Problem 9653

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

Problem 9654

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

Problem 9655

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

Problem 9656

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (\left (\beta +\alpha +1\right ) x^{2}-\left (\alpha +\beta +1+a \left (\gamma +\delta \right )-\delta \right ) x +a \gamma \right ) y^{\prime }}{x \left (x -1\right ) \left (x -a \right )}+\frac {\left (\alpha \beta x -q \right ) y}{x \left (x -1\right ) \left (x -a \right )}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {HeunG}\left (a , q , \alpha , \beta , \gamma , \delta , x\right )+c_{2} x^{1-\gamma } \operatorname {HeunG}\left (a , q -\left (-1+\gamma \right ) \left (\delta \left (a -1\right )+\alpha +\beta -\gamma +1\right ), \beta +1-\gamma , \alpha +1-\gamma , -\gamma +2, \delta , x\right ) \]

Problem 9657

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (A \,x^{2}+B x +C \right ) y^{\prime }}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}+\frac {\left (\operatorname {DD} x +E \right ) y}{\left (x -a \right ) \left (x -b \right ) \left (x -c \right )}=0} \]
program solution
Maple solution	\[ \text {Expression too large to display} \]

Problem 9658

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

Problem 9659

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

Problem 9660

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (-1+3 x \right ) y^{\prime }}{2 x \left (x -1\right )}-\frac {v \left (v +1\right ) y}{4 x^{2}}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {\left (\Gamma \left (v +\frac {1}{2}\right )^{2} c_{2} \left (v +\frac {1}{2}\right ) \operatorname {LegendreP}\left (-\frac {1}{2}, -v -\frac {1}{2}, \frac {-x -1}{x -1}\right )+\sec \left (\pi v \right ) \operatorname {LegendreP}\left (-\frac {1}{2}, v +\frac {1}{2}, \frac {-x -1}{x -1}\right ) \pi c_{1} \right ) x^{\frac {1}{4}}}{\sqrt {1-x}\, \Gamma \left (v +\frac {1}{2}\right )} \]

Problem 9661

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

Problem 9662

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (-1+3 x \right ) y^{\prime }}{2 x \left (x -1\right )}+\frac {\left (x a +b \right ) y}{4 x \left (x -1\right )^{2}}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {LegendreP}\left (\frac {\sqrt {-4 a +1}}{2}-\frac {1}{2}, \sqrt {-a -b}, \sqrt {x}\right )+c_{2} \operatorname {LegendreQ}\left (\frac {\sqrt {-4 a +1}}{2}-\frac {1}{2}, \sqrt {-a -b}, \sqrt {x}\right ) \]

Problem 9663

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

Problem 9664

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

Problem 9665

ODE	\[ \boxed {y^{\prime \prime }-\frac {\left (6 x -1\right ) y^{\prime }}{3 x \left (x -2\right )}-\frac {y}{3 x^{2} \left (x -2\right )}=0} \]
program solution	\[ y = c_{1} x^{\frac {1}{6}} \left (x -2\right )^{\frac {17}{6}}+c_{2} \left (-\frac {54}{935} x^{3}+\frac {18}{55} x^{2}-\frac {3}{5} x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x \left (18 x^{2}-102 x +187\right )+c_{2} \left (x -2\right )^{\frac {17}{6}} x^{\frac {1}{6}} \]

Problem 9666

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (a \left (2+b \right ) x^{2}+\left (c -d +1\right ) x \right ) y^{\prime }}{\left (x a +1\right ) x^{2}}+\frac {\left (a b x -c d \right ) y}{\left (x a +1\right ) x^{2}}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \left (x^{-c} \operatorname {hypergeom}\left (\left [-d , 1-b -d \right ], \left [1-d -c \right ], -a x \right ) c_{2} +x^{d} \operatorname {hypergeom}\left (\left [c , 1-b +c \right ], \left [1+d +c \right ], -a x \right ) c_{1} \right ) \left (a x +1\right )^{-b +c -d} \]

Problem 9667

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

Problem 9668

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (2 x a +b \right ) y^{\prime }}{x \left (x a +b \right )}+\frac {\left (a v x -b \right ) y}{\left (x a +b \right ) x^{2}}=A x} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {x^{-\frac {\sqrt {1-4 v}}{2}} a^{2} c_{1} \left (v +6\right ) \left (2+v \right ) \left (v +12\right ) \operatorname {hypergeom}\left (\left [-\frac {1}{2}+\frac {\sqrt {1-4 v}}{2}, \frac {3}{2}+\frac {\sqrt {1-4 v}}{2}\right ], \left [1+\sqrt {1-4 v}\right ], -\frac {b}{a x}\right )-3 b^{2} A \left (v +4\right ) x^{\frac {3}{2}}+\left (2+v \right ) \left (A b \left (v +4\right ) x^{\frac {5}{2}}+\left (A \,x^{\frac {7}{2}}+\left (v +12\right ) x^{\frac {\sqrt {1-4 v}}{2}} \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {\sqrt {1-4 v}}{2}, \frac {3}{2}-\frac {\sqrt {1-4 v}}{2}\right ], \left [1-\sqrt {1-4 v}\right ], -\frac {b}{a x}\right ) c_{2} \right ) \left (v +6\right ) a \right ) a}{\sqrt {x}\, a^{2} \left (v +6\right ) \left (2+v \right ) \left (v +12\right )} \]

Problem 9669

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

Problem 9670

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

Problem 9671

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

Problem 9672

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

Problem 9673

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

Problem 9674

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

Problem 9675

ODE	\[ \boxed {y^{\prime \prime }+\frac {y^{\prime }}{x}+\frac {\left (b \,x^{2}+a \left (x^{4}+1\right )\right ) y}{x^{4}}=0} \]
program solution	\[ y = c_{1} \operatorname {BesselJ}\left (2 \sqrt {-b}, 2 \sqrt {x}\right )+c_{2} \operatorname {BesselY}\left (2 \sqrt {-b}, 2 \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \operatorname {HeunD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{x^{2}-1}\right ) \left (c_{1} +c_{2} \left (\int \frac {1}{x \operatorname {HeunD}\left (0, 2 a +b , 0, 2 a -b , \frac {x^{2}+1}{x^{2}-1}\right )^{2}}d x \right )\right ) \]

Problem 9676

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

Problem 9677

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

Problem 9678

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

Problem 9679

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

Problem 9680

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

Problem 9681

ODE	\[ \boxed {y^{\prime \prime }-\frac {\left (2 x^{2}-1\right ) y^{\prime }}{x^{3}}+\frac {2 y}{x^{4}}=0} \]
program solution	\[ y = \frac {c_{1} \left (5 x^{2}-1\right )}{5 x^{2}}+\frac {c_{2} \left (25 x^{2}-5\right ) \left (\int \frac {x^{6} {\mathrm e}^{\frac {1}{2 x^{2}}}}{\left (5 x^{2}-1\right )^{2}}d x \right )}{x^{2}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{5} \operatorname {hypergeom}\left (\left [-\frac {5}{2}\right ], \left [-\frac {1}{2}\right ], \frac {1}{2 x^{2}}\right )+5 c_{1} x^{2}-c_{1}}{x^{2}} \]

Problem 9682

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

Problem 9683

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (2 x^{2}+1\right ) y^{\prime }}{x \left (x^{2}+1\right )}+\frac {\left (-v \left (v +1\right ) x^{2}-n^{2}\right ) y}{x^{2} \left (x^{2}+1\right )}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \operatorname {LegendreQ}\left (v , n , \sqrt {x^{2}+1}\right ) c_{2} +\operatorname {LegendreP}\left (v , n , \sqrt {x^{2}+1}\right ) c_{1} \]

Problem 9684

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (x^{2} a +a -1\right ) y^{\prime }}{x \left (x^{2}+1\right )}+\frac {\left (b \,x^{2}+c \right ) y}{x^{2} \left (x^{2}+1\right )}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = x^{1-\frac {a}{2}} \left (\operatorname {LegendreP}\left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {\sqrt {a^{2}-4 a -4 c +4}}{2}, \sqrt {x^{2}+1}\right ) c_{1} +\operatorname {LegendreQ}\left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {\sqrt {a^{2}-4 a -4 c +4}}{2}, \sqrt {x^{2}+1}\right ) c_{2} \right ) \]

Problem 9685

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

Problem 9686

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

Problem 9687

ODE	\[ \boxed {y^{\prime \prime }+\frac {2 x y^{\prime }}{x^{2}-1}-\frac {v \left (v +1\right ) y}{x^{2}}=0} \]
program solution	\[ y = \frac {c_{1} \operatorname {BesselJ}\left (-v -\frac {1}{2}, \frac {\sqrt {v^{2}+v}}{x}\right )}{\sqrt {x}}+\frac {c_{2} \operatorname {BesselY}\left (-v -\frac {1}{2}, \frac {\sqrt {v^{2}+v}}{x}\right )}{\sqrt {x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} x^{v +1} \left (x^{2}\right )^{-\frac {v}{2}-\frac {1}{4}} \Gamma \left (v +\frac {1}{2}\right )^{2} \left (v +\frac {1}{2}\right ) \operatorname {LegendreP}\left (-\frac {1}{2}, -v -\frac {1}{2}, \frac {-x^{2}-1}{x^{2}-1}\right )+\sec \left (\pi v \right ) \pi \operatorname {LegendreP}\left (-\frac {1}{2}, v +\frac {1}{2}, \frac {-x^{2}-1}{x^{2}-1}\right ) \left (x^{2}\right )^{\frac {1}{4}+\frac {v}{2}} x^{-v} c_{1}}{\sqrt {-x^{2}+1}\, \Gamma \left (v +\frac {1}{2}\right )} \]

Problem 9688

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

Problem 9689

ODE	\[ \boxed {y^{\prime \prime } x^{2} \left (x^{2}-1\right )-2 x^{3} y^{\prime }-\left (\left (a -n \right ) \left (a +n +1\right ) x^{2} \left (x^{2}-1\right )+2 x^{2} a +n \left (1+n \right ) \left (x^{2}-1\right )\right ) y=0} \]
program solution	\[ y = c_{1} x^{\frac {3}{2}} \operatorname {BesselJ}\left (\sqrt {-4 a^{2}+8 n^{2}+4 a +8 n +9}, 2 \sqrt {x}\right )+c_{2} x^{\frac {3}{2}} \operatorname {BesselY}\left (\sqrt {-4 a^{2}+8 n^{2}+4 a +8 n +9}, 2 \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = c_{1} x^{-n} \operatorname {HeunC}\left (0, -n -\frac {1}{2}, -2, -\frac {1}{4} a^{2}+\frac {1}{4} n^{2}-\frac {1}{4} a +\frac {1}{4} n , -\frac {1}{4} n^{2}-\frac {1}{4} n +\frac {3}{4}+\frac {1}{4} a^{2}-\frac {1}{4} a , x^{2}\right )+c_{2} x^{1+n} \operatorname {HeunC}\left (0, n +\frac {1}{2}, -2, -\frac {1}{4} a^{2}+\frac {1}{4} n^{2}-\frac {1}{4} a +\frac {1}{4} n , -\frac {1}{4} n^{2}-\frac {1}{4} n +\frac {3}{4}+\frac {1}{4} a^{2}-\frac {1}{4} a , x^{2}\right ) \]

Problem 9690

ODE	\[ \boxed {y^{\prime \prime }+\frac {\left (x^{2} a +a -2\right ) y^{\prime }}{x \left (x^{2}-1\right )}+\frac {b y}{x^{2}}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = x^{\frac {a}{2}-\frac {1}{2}} \left (-x^{2}+1\right )^{\frac {1}{2}+\frac {a}{2}} \left (x^{2}-1\right )^{-a} \left (x^{\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}} \left (x^{2}\right )^{-\frac {\sqrt {a^{2}-2 a -4 b +1}}{4}} \operatorname {LegendreP}\left (\frac {a}{2}-\frac {3}{2}, -\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {-x^{2}-1}{x^{2}-1}\right ) c_{1} \Gamma \left (1+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}\right )+\frac {\csc \left (\frac {\pi \sqrt {a^{2}-2 a -4 b +1}}{2}\right ) x^{-\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}} \left (x^{2}\right )^{\frac {\sqrt {a^{2}-2 a -4 b +1}}{4}} \operatorname {LegendreP}\left (\frac {a}{2}-\frac {3}{2}, \frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {-x^{2}-1}{x^{2}-1}\right ) \sqrt {a^{2}-2 a -4 b +1}\, \pi c_{2}}{2 \Gamma \left (1+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}\right )}\right ) \]

Problem 9691

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

Problem 9692

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

Problem 9693

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

Problem 9694

ODE	\[ \boxed {y^{\prime \prime }+\frac {2 x y^{\prime }}{x^{2}+1}+\frac {\left (a^{2} \left (x^{2}+1\right )^{2}-n \left (1+n \right ) \left (x^{2}+1\right )+m^{2}\right ) y}{\left (x^{2}+1\right )^{2}}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \left (\operatorname {HeunC}\left (0, \frac {1}{2}, m , -\frac {a^{2}}{4}, \frac {1}{4}+\frac {1}{4} a^{2}+\frac {1}{4} m^{2}-\frac {1}{4} n^{2}-\frac {1}{4} n , -x^{2}\right ) c_{2} x +\operatorname {HeunC}\left (0, -\frac {1}{2}, m , -\frac {a^{2}}{4}, \frac {1}{4}+\frac {1}{4} a^{2}+\frac {1}{4} m^{2}-\frac {1}{4} n^{2}-\frac {1}{4} n , -x^{2}\right ) c_{1} \right ) \left (x^{2}+1\right )^{\frac {m}{2}} \]

Problem 9695

ODE	\[ \boxed {y^{\prime \prime }+\frac {a x y^{\prime }}{x^{2}+1}+\frac {b y}{\left (x^{2}+1\right )^{2}}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \left (x^{2}+1\right )^{\frac {1}{2}-\frac {a}{4}} \left (\operatorname {LegendreP}\left (\frac {a}{2}-1, \frac {\sqrt {a^{2}-4 a +4 b +4}}{2}, i x \right ) c_{1} +\operatorname {LegendreQ}\left (\frac {a}{2}-1, \frac {\sqrt {a^{2}-4 a +4 b +4}}{2}, i x \right ) c_{2} \right ) \]

Problem 9696

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

Problem 9697

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

Problem 9698

ODE	\[ \boxed {y^{\prime \prime }+\frac {2 x y^{\prime }}{x^{2}-1}+\frac {\left (-a^{2}-\lambda \left (x^{2}-1\right )\right ) y}{\left (x^{2}-1\right )^{2}}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {LegendreP}\left (\frac {\sqrt {1+4 \lambda }}{2}-\frac {1}{2}, a , x\right )+c_{2} \operatorname {LegendreQ}\left (\frac {\sqrt {1+4 \lambda }}{2}-\frac {1}{2}, a , x\right ) \]

Problem 9699

ODE	\[ \boxed {y^{\prime \prime }+\frac {2 x y^{\prime }}{x^{2}-1}+\frac {\left (\left (x^{2}-1\right ) \left (x^{2} a +b x +c \right )-k^{2}\right ) y}{\left (x^{2}-1\right )^{2}}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\sqrt {-a}\, x} \left (\left (x^{2}-1\right )^{\frac {k}{2}} \operatorname {HeunC}\left (4 \sqrt {-a}, k , k , 2 b , \frac {k^{2}}{2}+a -b +c , \frac {x}{2}+\frac {1}{2}\right ) c_{1} +\left (x -1\right )^{\frac {k}{2}} \left (x +1\right )^{-\frac {k}{2}} \operatorname {HeunC}\left (4 \sqrt {-a}, -k , k , 2 b , \frac {k^{2}}{2}+a -b +c , \frac {x}{2}+\frac {1}{2}\right ) c_{2} \right ) \]

Problem 9700

ODE	\[ \boxed {y^{\prime \prime }+\frac {2 x y^{\prime }}{x^{2}-1}+\frac {\left (-a^{2} \left (x^{2}-1\right )^{2}-n \left (1+n \right ) \left (x^{2}-1\right )-m^{2}\right ) y}{\left (x^{2}-1\right )^{2}}=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \left (\operatorname {HeunC}\left (0, \frac {1}{2}, m , -\frac {a^{2}}{4}, \frac {1}{4}+\frac {1}{4} a^{2}+\frac {1}{4} m^{2}-\frac {1}{4} n^{2}-\frac {1}{4} n , x^{2}\right ) c_{2} x +\operatorname {HeunC}\left (0, -\frac {1}{2}, m , -\frac {a^{2}}{4}, \frac {1}{4}+\frac {1}{4} a^{2}+\frac {1}{4} m^{2}-\frac {1}{4} n^{2}-\frac {1}{4} n , x^{2}\right ) c_{1} \right ) \left (x^{2}-1\right )^{\frac {m}{2}} \]