2.17.96 Problems 9501 to 9600

Problem 9501

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

Problem 9502

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

Problem 9503

ODE	\[ \boxed {x^{2} y^{\prime \prime }-2 y^{\prime } x +2 y=x^{5} \ln \left (x \right )} \]
program solution	\[ y = x \left (c_{2} x +c_{1} \right )+\frac {x^{5} \left (-7+12 \ln \left (x \right )\right )}{144} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {x^{5} \ln \left (x \right )}{12}-\frac {7 x^{5}}{144}+c_{2} x^{2}+c_{1} x \]

Problem 9504

ODE	\[ \boxed {x^{2} y^{\prime \prime }-2 y^{\prime } x -4 y=x \sin \left (x \right )+\left (x^{2} a +12 a +4\right ) \cos \left (x \right )} \]
program solution	\[ y = \frac {\left (-10 a -5\right ) \sin \left (x \right )+5 c_{2} x^{5}-5 x a \cos \left (x \right )-c_{1}}{5 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (-2 a -1\right ) \sin \left (x \right )+x^{5} c_{2} -x a \cos \left (x \right )+c_{1}}{x} \]

Problem 9505

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

Problem 9506

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

Problem 9507

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

Problem 9508

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

Problem 9509

ODE	\[ \boxed {x^{2} y^{\prime \prime }+3 y^{\prime } x +\left (-v^{2}+x^{2}+1\right ) y=f \left (x \right )} \]
program solution	\[ y = \frac {c_{1} \operatorname {BesselJ}\left (v , x\right )}{x}+\frac {c_{2} \operatorname {BesselY}\left (v , x\right )}{x}-\frac {\pi \left (\left (\int _{0}^{x}\operatorname {BesselY}\left (v , \alpha \right ) f \left (\alpha \right )d \alpha \right ) \operatorname {BesselJ}\left (v , x\right )-\left (\int _{0}^{x}\operatorname {BesselJ}\left (v , \alpha \right ) f \left (\alpha \right )d \alpha \right ) \operatorname {BesselY}\left (v , x\right )\right )}{2 x} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\pi \left (\int \operatorname {BesselJ}\left (v , x\right ) f \left (x \right )d x \right ) \operatorname {BesselY}\left (v , x\right )-\pi \left (\int \operatorname {BesselY}\left (v , x\right ) f \left (x \right )d x \right ) \operatorname {BesselJ}\left (v , x\right )+2 \operatorname {BesselY}\left (v , x\right ) c_{1} +2 \operatorname {BesselJ}\left (v , x\right ) c_{2}}{2 x} \]

Problem 9510

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

Problem 9511

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

Problem 9512

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

Problem 9513

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

Problem 9514

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

Problem 9515

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

Problem 9516

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

Problem 9517

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

Problem 9518

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

Problem 9519

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

Problem 9520

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

Problem 9521

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (x^{2}-1\right ) y^{\prime }-y=0} \]
program solution	\[ y = c_{1} {\mathrm e}^{-x}+c_{2} {\mathrm e}^{-x} \left (\int {\mathrm e}^{\frac {\left (x -1\right ) \left (x +1\right )}{x}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (\operatorname {HeunD}\left (-4, 3, -8, 5, \frac {x -1}{x +1}\right ) {\mathrm e}^{-\frac {1}{x}} c_{2} +{\mathrm e}^{-x} \operatorname {HeunD}\left (4, 3, -8, 5, \frac {x -1}{x +1}\right ) c_{1} \right ) \sqrt {x} \]

Problem 9522

ODE	\[ \boxed {x^{2} y^{\prime \prime }+x \left (x +1\right ) y^{\prime }+\left (x -9\right ) y=0} \]
program solution	\[ y = \frac {c_{1} \left (x^{2}-8 x +20\right )}{x^{3}}-\frac {c_{2} \left (x^{3}+9 x^{2}+36 x +60\right ) {\mathrm e}^{-x}}{x^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{2} {\mathrm e}^{-x} \left (x^{3}+9 x^{2}+36 x +60\right )+c_{1} \left (x^{2}-8 x +20\right )}{x^{3}} \]

Problem 9523

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

Problem 9524

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

Problem 9525

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

Problem 9526

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

Problem 9527

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

Problem 9528

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

Problem 9529

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

Problem 9530

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

Problem 9531

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

Problem 9532

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

Problem 9533

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

Problem 9534

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

Problem 9535

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

Problem 9536

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (x a +b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{-\frac {a x}{2}} x^{-\frac {b}{2}} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {a b -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2}, \sqrt {a^{2}-4 \operatorname {a1}}\, x \right )+c_{2} \operatorname {WhittakerW}\left (-\frac {a b -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2}, \sqrt {a^{2}-4 \operatorname {a1}}\, x \right )\right ) \]

Problem 9537

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

Problem 9538

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

Problem 9539

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

Problem 9540

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

Problem 9541

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

Problem 9542

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

Problem 9543

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

Problem 9544

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (x^{n} a +b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2 n}+\operatorname {b1} \,x^{n}+\operatorname {c1} \right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = x^{-\frac {b}{2}-\frac {n}{2}+\frac {1}{2}} {\mathrm e}^{-\frac {a \,x^{n}}{2 n}} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {\left (b +n -1\right ) a -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}\, n}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2 n}, \frac {\sqrt {a^{2}-4 \operatorname {a1}}\, x^{n}}{n}\right )+c_{2} \operatorname {WhittakerW}\left (-\frac {\left (b +n -1\right ) a -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}\, n}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2 n}, \frac {\sqrt {a^{2}-4 \operatorname {a1}}\, x^{n}}{n}\right )\right ) \]

Problem 9545

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{\operatorname {a1}}+b \right ) x y^{\prime }+\left (A \,x^{2 \operatorname {a1}}+B \,x^{\operatorname {a1}}+C \,x^{\operatorname {b1}}+\operatorname {DD} \right ) y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 9546

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

Problem 9547

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (2 x^{2} \cot \left (x \right )+x \right ) y^{\prime }+\left (\cot \left (x \right ) x +a \right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \csc \left (x \right ) \left (c_{1} \operatorname {BesselJ}\left (i \sqrt {a}, x\right )+c_{2} \operatorname {BesselY}\left (i \sqrt {a}, x\right )\right ) \]

Problem 9548

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

Problem 9549

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

Problem 9550

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

Problem 9551

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

Problem 9552

ODE	\[ \boxed {\left (x^{2}+1\right ) y^{\prime \prime }+y^{\prime } x -9 y=0} \]
program solution	\[ y = \frac {c_{1} \left (4 x^{2}+1\right ) \sqrt {x^{2}+1}}{4}+c_{2} \left (\frac {16}{3} x^{3}+4 x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \sqrt {x^{2}+1}\, \left (4 x^{2}+1\right ) c_{2} +4 c_{1} x^{3}+3 c_{1} x \]

Problem 9553

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

Problem 9554

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

Problem 9555

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

Problem 9556

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

Problem 9557

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

Problem 9558

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

Problem 9559

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

Problem 9560

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

Problem 9561

ODE	\[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }-n \left (1+n \right ) y=-\frac {\partial }{\partial x}\operatorname {LegendreP}\left (n , x\right )} \]
program solution
Maple solution	\[ y \left (x \right ) = 3 \left (x +1\right ) \left (x -1\right ) \left (-\operatorname {hypergeom}\left (\left [\frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right ) \left (1+n \right ) \left (\int \frac {x \operatorname {hypergeom}\left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) \left (x \operatorname {LegendreP}\left (n , x\right )-\operatorname {LegendreP}\left (1+n , x\right )\right )}{3 \left (x +1\right )^{3} \left (x -1\right )^{3} \left (\left (\operatorname {hypergeom}\left (\left [\frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right )+\left (n^{2}+n -2\right ) x^{2} \operatorname {hypergeom}\left (\left [\frac {n}{2}+2, \frac {3}{2}-\frac {n}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right )\right ) \operatorname {hypergeom}\left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right )-\frac {\operatorname {hypergeom}\left (\left [\frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right ) \operatorname {hypergeom}\left (\left [-\frac {n}{2}+2, \frac {n}{2}+\frac {5}{2}\right ], \left [\frac {5}{2}\right ], x^{2}\right ) x^{2} \left (n +3\right ) \left (n -2\right )}{3}\right )}d x \right )+\operatorname {hypergeom}\left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) x \left (1+n \right ) \left (\int \frac {\operatorname {hypergeom}\left (\left [\frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right ) \left (x \operatorname {LegendreP}\left (n , x\right )-\operatorname {LegendreP}\left (1+n , x\right )\right )}{3 \left (x +1\right )^{3} \left (x -1\right )^{3} \left (\left (\operatorname {hypergeom}\left (\left [\frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right )+\left (n^{2}+n -2\right ) x^{2} \operatorname {hypergeom}\left (\left [\frac {n}{2}+2, \frac {3}{2}-\frac {n}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right )\right ) \operatorname {hypergeom}\left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right )-\frac {\operatorname {hypergeom}\left (\left [\frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right ) \operatorname {hypergeom}\left (\left [-\frac {n}{2}+2, \frac {n}{2}+\frac {5}{2}\right ], \left [\frac {5}{2}\right ], x^{2}\right ) x^{2} \left (n +3\right ) \left (n -2\right )}{3}\right )}d x \right )-\frac {\operatorname {hypergeom}\left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {3}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) c_{1} x}{3}-\frac {\operatorname {hypergeom}\left (\left [\frac {n}{2}+1, -\frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right ) c_{2}}{3}\right ) \]

Problem 9562

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

Problem 9563

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

Problem 9564

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

Problem 9565

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

Problem 9566

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

Problem 9567

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

Problem 9568

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

Problem 9569

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

Problem 9570

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

Problem 9571

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

Problem 9572

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

Problem 9573

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

Problem 9574

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

Problem 9575

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

Problem 9576

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

Problem 9577

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

Problem 9578

ODE	\[ \boxed {\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+8 y^{\prime } x +12 y=0} \]
program solution	\[ y = -\frac {c_{1}}{\left (a -x \right )^{3}}+\frac {c_{2} \left (a^{2}+3 x^{2}\right )}{3 \left (a -x \right )^{3} \left (x +a \right )^{3}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {3 a^{2} c_{2} x +c_{2} x^{3}+a^{2} c_{1} +3 c_{1} x^{2}}{\left (a +x \right )^{3} \left (-x +a \right )^{3}} \]

Problem 9579

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

Problem 9580

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

Problem 9581

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

Problem 9582

ODE	\[ \boxed {\left (x^{2}+x -2\right ) y^{\prime \prime }+\left (x^{2}-x \right ) y^{\prime }-\left (6 x^{2}+7 x \right ) y=0} \]
program solution	\[ y = c_{1} \left (x -1\right ) {\mathrm e}^{2 x}+39 c_{2} {\mathrm e}^{-3 x} \left ({\mathrm e}^{5 x -5} \left (x -1\right ) \operatorname {expIntegral}_{1}\left (5 x -5\right )-\frac {x}{195}-\frac {44}{195}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (195 \,{\mathrm e}^{5 x -5} c_{2} \left (x -1\right ) \operatorname {expIntegral}_{1}\left (5 x -5\right )+\left (x -1\right ) c_{1} {\mathrm e}^{5 x}-c_{2} \left (x +44\right )\right ) {\mathrm e}^{-3 x} \]

Problem 9583

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

Problem 9584

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

Problem 9585

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

Problem 9586

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

Problem 9587

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

Problem 9588

ODE	\[ \boxed {x \left (x -1\right ) y^{\prime \prime }+\left (\left (\operatorname {a1} +\operatorname {b1} +1\right ) x -\operatorname {d1} \right ) y^{\prime }=-\operatorname {a1} \operatorname {b1} \operatorname {d1}} \]
program solution	\[ y = \int \left (-\left (-1\right )^{-\operatorname {a1} -\operatorname {b1} +\operatorname {d1}} \operatorname {hypergeom}\left (\left [\operatorname {d1} , -\operatorname {a1} -\operatorname {b1} +\operatorname {d1} \right ], \left [\operatorname {d1} +1\right ], x\right ) \operatorname {a1} \operatorname {b1} +x^{-\operatorname {d1}} c_{1} \right ) \left (x -1\right )^{-\operatorname {a1} -\operatorname {b1} +\operatorname {d1} -1}d x +c_{2} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \int \left (x -1\right )^{-\operatorname {a1} -\operatorname {b1} -1+\operatorname {d1}} \left (-\operatorname {a1} \operatorname {b1} \operatorname {signum}\left (x -1\right )^{\operatorname {a1} +\operatorname {b1} -\operatorname {d1}} \left (-\operatorname {signum}\left (x -1\right )\right )^{-\operatorname {a1} -\operatorname {b1} +\operatorname {d1}} \operatorname {hypergeom}\left (\left [\operatorname {d1} , -\operatorname {a1} -\operatorname {b1} +\operatorname {d1} \right ], \left [1+\operatorname {d1} \right ], x\right )+x^{-\operatorname {d1}} c_{1} \right )d x +c_{2} \]

Problem 9589

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

Problem 9590

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

Problem 9591

ODE	\[ \boxed {x \left (x +3\right ) y^{\prime \prime }+\left (-1+3 x \right ) y^{\prime }+y=\left (20 x +30\right ) \left (x^{2}+3 x \right )^{\frac {7}{3}}} \]
program solution	\[ y = \frac {-\frac {9 c_{1} 3^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [-\frac {4}{3}, -\frac {4}{3}\right ], \left [-\frac {1}{3}\right ], -\frac {x}{3}\right )}{4}+\frac {9 \left (x^{\frac {16}{3}}-\frac {243 x^{\frac {4}{3}}}{17}+\frac {216 x^{\frac {7}{3}}}{17}+\frac {378 x^{\frac {10}{3}}}{17}+\frac {144 x^{\frac {13}{3}}}{17}\right ) \left (x +3\right )^{\frac {8}{3}}}{20}+\left (c_{2} +\frac {19683 \,3^{\frac {2}{3}}}{340}\right ) x^{\frac {4}{3}}}{\left (x +3\right )^{\frac {7}{3}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (c_{2} +\int \frac {3 \left (x^{3} \left (x +3\right )^{3} \left (x \left (x +3\right )\right )^{\frac {1}{3}}+\frac {c_{1}}{3}\right ) \left (x +3\right )^{\frac {4}{3}}}{x^{\frac {7}{3}}}d x \right ) x^{\frac {4}{3}}}{\left (x +3\right )^{\frac {7}{3}}} \]

Problem 9592

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

Problem 9593

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

Problem 9594

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

Problem 9595

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

Problem 9596

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

Problem 9597

ODE	\[ \boxed {2 x \left (x -1\right ) y^{\prime \prime }+\left (\left (2 v +5\right ) x -2 v -3\right ) y^{\prime }+\left (v +1\right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {x^{-\frac {v}{2}-\frac {1}{4}} \left (c_{1} \Gamma \left (v +\frac {1}{2}\right )^{2} \left (v +\frac {1}{2}\right ) \operatorname {LegendreP}\left (-\frac {1}{2}, -v -\frac {1}{2}, \frac {-x -1}{x -1}\right )+\sec \left (\pi v \right ) \pi \operatorname {LegendreP}\left (-\frac {1}{2}, v +\frac {1}{2}, \frac {-x -1}{x -1}\right ) c_{2} \right )}{\sqrt {1-x}\, \Gamma \left (v +\frac {1}{2}\right )} \]

Problem 9598

ODE	\[ \boxed {\left (2 x^{2}+6 x +4\right ) y^{\prime \prime }+\left (10 x^{2}+21 x +8\right ) y^{\prime }+\left (12 x^{2}+17 x +8\right ) y=0} \]
program solution	\[ y = c_{1} \left (x +2\right )^{4} {\mathrm e}^{-3 x}+c_{2} \left (x +2\right )^{4} {\mathrm e}^{-3 x} \left (\int \frac {{\mathrm e}^{x} \left (x +1\right )^{\frac {3}{2}}}{\left (x +2\right )^{5}}d x \right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \left (c_{2} \left (x +1\right )^{\frac {5}{2}} \operatorname {HeunC}\left (-1, \frac {5}{2}, 4, -\frac {7}{4}, \frac {7}{2}, -x -1\right )+c_{1} \operatorname {HeunC}\left (-1, -\frac {5}{2}, 4, -\frac {7}{4}, \frac {7}{2}, -x -1\right )\right ) {\mathrm e}^{-2 x} \left (x +2\right )^{4} \]

Problem 9599

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

Problem 9600

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