2.17.110 Problems 10901 to 11000

Problem 10901

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

Problem 10902

ODE	\[ \boxed {x y^{\prime \prime }+\left (a b \,x^{2}+b -5\right ) y^{\prime }+2 a^{2} \left (-2+b \right ) x^{3} y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {\left (-3 \operatorname {KummerU}\left (\frac {b}{2}+1, -2+\frac {b}{2}, \frac {a \left (b -4\right ) x^{2}}{2}\right ) c_{2} b +\left (a \left (b -4\right ) x^{2}+b +4\right ) c_{2} \operatorname {KummerU}\left (\frac {b}{2}, -2+\frac {b}{2}, \frac {a \left (b -4\right ) x^{2}}{2}\right )+2 c_{1} {\mathrm e}^{\frac {a \left (b -4\right ) x^{2}}{2}} \left (a \,x^{2}+1\right )\right ) {\mathrm e}^{-\frac {a \left (b -2\right ) x^{2}}{2}}}{2} \]

Problem 10903

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

Problem 10904

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

Problem 10905

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

Problem 10906

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

Problem 10907

ODE	\[ \boxed {x y^{\prime \prime }+\left (a \,x^{2}+b x +c \right ) y^{\prime }+\left (A \,x^{2}+B x +\operatorname {C0} \right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = {\mathrm e}^{\frac {x \left (-a^{2} x -2 a b +2 A \right )}{2 a}} \left (x^{-c +1} \operatorname {HeunB}\left (-c +1, -\frac {\sqrt {2}\, \left (-a b +2 A \right )}{a^{\frac {3}{2}}}, \frac {\left (-c -1\right ) a^{3}+2 B \,a^{2}-2 A a b +2 A^{2}}{a^{3}}, \frac {\left (b c -2 \operatorname {C0} \right ) \sqrt {2}}{\sqrt {a}}, \frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right ) c_{2} +\operatorname {HeunB}\left (c -1, -\frac {\sqrt {2}\, \left (-a b +2 A \right )}{a^{\frac {3}{2}}}, \frac {\left (-c -1\right ) a^{3}+2 B \,a^{2}-2 A a b +2 A^{2}}{a^{3}}, \frac {\left (b c -2 \operatorname {C0} \right ) \sqrt {2}}{\sqrt {a}}, \frac {\sqrt {2}\, \sqrt {a}\, x}{2}\right ) c_{1} \right ) \]

Problem 10908

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

Problem 10909

ODE	\[ \boxed {x y^{\prime \prime }+\left (a \,x^{3}+b \right ) y^{\prime }+a \left (b -1\right ) x^{2} y=0} \]
program solution	\[ y = \left (3^{-\frac {4}{3}+\frac {b}{3}} a^{-\frac {b}{3}+\frac {1}{3}} c_{1} \left (\frac {3^{-\frac {b}{6}+\frac {8}{3}} x^{2+b} a^{\frac {2}{3}+\frac {b}{3}} \left (a \,x^{3}\right )^{-\frac {1}{3}-\frac {b}{6}} {\mathrm e}^{-\frac {a \,x^{3}}{6}} \operatorname {WhittakerM}\left (\frac {1}{3}+\frac {b}{6}, \frac {b}{6}+\frac {5}{6}, \frac {a \,x^{3}}{3}\right )}{\left (b -1\right ) \left (2+b \right ) \left (5+b \right )}+\frac {3^{-\frac {b}{6}+\frac {8}{3}} x^{-4+b} a^{-\frac {4}{3}+\frac {b}{3}} \left (a \,x^{3}+b +2\right ) \left (a \,x^{3}\right )^{-\frac {1}{3}-\frac {b}{6}} {\mathrm e}^{-\frac {a \,x^{3}}{6}} \operatorname {WhittakerM}\left (\frac {4}{3}+\frac {b}{6}, \frac {b}{6}+\frac {5}{6}, \frac {a \,x^{3}}{3}\right )}{\left (b -1\right ) \left (2+b \right )}\right )+c_{2} \right ) x^{-b +1} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {9 c_{2} a^{2} x^{-\frac {b}{2}+3} {\mathrm e}^{-\frac {a \,x^{3}}{6}} \operatorname {WhittakerM}\left (\frac {1}{3}+\frac {b}{6}, \frac {b}{6}+\frac {5}{6}, \frac {a \,x^{3}}{3}\right )+\left (a \,x^{-\frac {b}{2}+3}+x^{-\frac {b}{2}} \left (b +2\right )\right ) c_{2} {\mathrm e}^{-\frac {a \,x^{3}}{3}} a 3^{-\frac {b}{6}+\frac {2}{3}} \left (b +5\right ) \left (a \,x^{3}\right )^{\frac {1}{3}+\frac {b}{6}}+9 c_{1} x^{-b +2}}{9 x} \]

Problem 10910

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

Problem 10911

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

Problem 10912

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

Problem 10913

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

Problem 10914

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

Problem 10915

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

Problem 10916

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

Problem 10917

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

Problem 10918

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

Problem 10919

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

Problem 10920

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

Problem 10921

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

Problem 10922

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

Problem 10923

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

Problem 10924

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

Problem 10925

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

Problem 10926

ODE	\[ \boxed {x y^{\prime \prime }+\left (a \,x^{n}+b \,x^{m}+c \right ) y^{\prime }+\left (c -1\right ) \left (a \,x^{n -1}+b \,x^{m -1}\right ) y=0} \]
program solution	\[ y = \left (\int \frac {c_{1} x^{c} {\mathrm e}^{-\frac {b \,x^{m}}{m}} {\mathrm e}^{-\frac {a \,x^{n}}{n}}}{x^{2}}d x +c_{2} \right ) x^{-c +1} \] Verified OK.
Maple solution	\[ \text {No solution found} \]

Problem 10927

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

Problem 10928

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

Problem 10929

ODE	\[ \boxed {\left (a_{1} x +a_{0} \right ) y^{\prime \prime }+\left (b_{1} x +b_{0} \right ) y^{\prime }-m b_{1} y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \left (a_{1} x +a_{0} \right )^{\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}} {\mathrm e}^{-\frac {b_{1} x}{a_{1}}} \left (\operatorname {KummerM}\left (1+m , \frac {a_{0} b_{1} +2 a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}, \frac {b_{1} \left (a_{1} x +a_{0} \right )}{a_{1}^{2}}\right ) c_{1} +\operatorname {KummerU}\left (1+m , \frac {a_{0} b_{1} +2 a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}, \frac {b_{1} \left (a_{1} x +a_{0} \right )}{a_{1}^{2}}\right ) c_{2} \right ) \]

Problem 10930

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

Problem 10931

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

Problem 10932

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

Problem 10933

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

Problem 10934

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

Problem 10935

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

Problem 10936

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

Problem 10937

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

Problem 10938

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

Problem 10939

ODE	\[ \boxed {x^{2} y^{\prime \prime }-\left (a \,x^{3}+\frac {5}{16}\right ) y=0} \]
program solution	\[ y = \frac {c_{1} {\mathrm e}^{\frac {2 x \sqrt {a x}}{3}}}{x^{\frac {1}{4}}}-\frac {c_{2} x^{\frac {1}{4}} \left (-{\mathrm e}^{\frac {2 x \sqrt {a x}}{3}}+{\mathrm e}^{-\frac {2 x \sqrt {a x}}{3}}\right )}{2 \sqrt {a x}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {c_{1} \sinh \left (\frac {2 x^{\frac {3}{2}} \sqrt {a}}{3}\right )+c_{2} \cosh \left (\frac {2 x^{\frac {3}{2}} \sqrt {a}}{3}\right )}{x^{\frac {1}{4}}} \]

Problem 10940

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

Problem 10941

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

Problem 10942

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

Problem 10943

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

Problem 10944

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{3 n}+b \,x^{2 n}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y=0} \]
program solution	\[ y = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (n , 2 \sqrt {x}\right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (n , 2 \sqrt {x}\right ) \] Verified OK.
Maple solution	\[ y \left (x \right ) = \frac {\left (\frac {2 \,3^{\frac {5}{6}} \pi c_{2} \left (a \,x^{n}+b \right ) \operatorname {BesselI}\left (\frac {1}{3}, \frac {2 \sqrt {\frac {-x^{3 n} a^{3}-3 x^{2 n} a^{2} b -3 x^{n} a \,b^{2}-b^{3}}{n^{2} a^{2}}}}{3}\right )}{3}+c_{1} \operatorname {BesselI}\left (-\frac {1}{3}, \frac {2 \sqrt {\frac {-x^{3 n} a^{3}-3 x^{2 n} a^{2} b -3 x^{n} a \,b^{2}-b^{3}}{n^{2} a^{2}}}}{3}\right ) \Gamma \left (\frac {2}{3}\right )^{2} 3^{\frac {2}{3}} {\left (-\frac {\left (a \,x^{n}+b \right )^{3}}{a^{2} n^{2}}\right )}^{\frac {1}{3}}\right ) x^{-\frac {n}{2}+\frac {1}{2}}}{3 {\left (-\frac {\left (a \,x^{n}+b \right )^{3}}{a^{2} n^{2}}\right )}^{\frac {1}{6}} \Gamma \left (\frac {2}{3}\right )} \]

Problem 10945

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{2 n} \left (b \,x^{n}+c \right )^{m}+\frac {1}{4}-\frac {n^{2}}{4}\right ) y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10946

ODE	\[ \boxed {x^{2} y^{\prime \prime }+y^{\prime } a x +y b=0} \]
program solution	\[ y = c_{1} x^{-\frac {a}{2}+\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}}-\frac {c_{2} x^{-\frac {a}{2}+\frac {1}{2}-\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}}}{\sqrt {a^{2}-2 a -4 b +1}} \] 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 10947

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

Problem 10948

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

Problem 10949

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

Problem 10950

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

Problem 10951

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

Problem 10952

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

Problem 10953

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

Problem 10954

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

Problem 10955

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

Problem 10956

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

Problem 10957

ODE	\[ \boxed {x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+y c=0} \]
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 10958

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

Problem 10959

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

Problem 10960

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

Problem 10961

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

Problem 10962

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

Problem 10963

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

Problem 10964

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 10965

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

Problem 10966

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

Problem 10967

ODE	\[ \boxed {x^{2} y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (a b \,x^{n +2 m}-b^{2} x^{2+4 m}+a m \,x^{n -1}-m^{2}-m \right ) y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 10968

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

Problem 10969

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

Problem 10970

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

Problem 10971

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

Problem 10972

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

Problem 10973

ODE	\[ \boxed {\left (-a^{2}+x^{2}\right ) y^{\prime \prime }+y^{\prime } b -6 y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = -\frac {c_{1} \left (10 a^{2} b +24 a^{2} x -b^{3}-6 b^{2} x -18 x^{2} b -24 x^{3}\right )}{24}+c_{2} \left (a +x \right ) \left (a -x \right ) \left (b -4 x \right ) \left (\frac {a +x}{a -x}\right )^{\frac {b}{2 a}} \]

Problem 10974

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 10975

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

Problem 10976

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

Problem 10977

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

Problem 10978

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

Problem 10979

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

Problem 10980

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

Problem 10981

ODE	\[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+\left (2 a +1\right ) y^{\prime }-b \left (2 a +b \right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\frac {\sqrt {8 a b +4 b^{2}+1}}{2}, \frac {\sqrt {8 a b +4 b^{2}+1}}{2}-\frac {1}{2}\right ], \left [-a -\frac {1}{2}\right ], \frac {1}{2}+\frac {x}{2}\right )+c_{2} \left (\frac {1}{2}+\frac {x}{2}\right )^{a +\frac {3}{2}} \operatorname {hypergeom}\left (\left [1-\frac {\sqrt {8 a b +4 b^{2}+1}}{2}+a , \frac {\sqrt {8 a b +4 b^{2}+1}}{2}+1+a \right ], \left [\frac {5}{2}+a \right ], \frac {1}{2}+\frac {x}{2}\right ) \]

Problem 10982

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

Problem 10983

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

Problem 10984

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

Problem 10985

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

Problem 10986

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

Problem 10987

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

Problem 10988

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

Problem 10989

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

Problem 10990

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

Problem 10991

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

Problem 10992

ODE	\[ \boxed {\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (c \,x^{2}+d \right ) y^{\prime }+\lambda \left (\left (-a \lambda +c \right ) x^{2}+d -\lambda b \right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \left (-a x +\sqrt {-a b}\right )^{\frac {2 a^{2} b +\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{4 a^{2} b}} \left (c_{2} \left (a x +\sqrt {-a b}\right )^{-\frac {-2 a^{2} b +\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{4 a^{2} b}} \operatorname {HeunC}\left (\frac {\left (4 a \lambda -2 c \right ) \sqrt {-\frac {b}{a}}}{a}, -\frac {\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{2 a^{2} b}, \frac {\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{2 a^{2} b}, 0, \frac {\lambda d}{a}-\frac {b c \lambda }{a^{2}}+\frac {1}{2}-\frac {d^{2}}{8 a b}-\frac {c d}{4 a^{2}}+\frac {3 b \,c^{2}}{8 a^{3}}, \frac {a x}{2 \sqrt {-a b}}+\frac {1}{2}\right ) {\mathrm e}^{\frac {-i \pi \sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}+i \pi \sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-4 b \left (a^{2} \left (\frac {d}{\sqrt {b}\, \sqrt {a}}-\frac {\sqrt {b}\, c}{a^{\frac {3}{2}}}\right ) \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right )+\left (-2 a \lambda +c \right ) \sqrt {-a b}-2 a x \left (a \lambda -c \right )\right )}{8 a^{2} b}}+c_{1} \left (a x +\sqrt {-a b}\right )^{\frac {2 a^{2} b +\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{4 a^{2} b}} {\mathrm e}^{x \lambda +\frac {\sqrt {-a b}\, \lambda }{a}-\frac {c x}{a}-\frac {\sqrt {-a b}\, c}{2 a^{2}}-\frac {\arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) d}{2 \sqrt {a}\, \sqrt {b}}+\frac {\sqrt {b}\, \arctan \left (\frac {\sqrt {a}\, x}{\sqrt {b}}\right ) c}{2 a^{\frac {3}{2}}}} \operatorname {HeunC}\left (\frac {\left (4 a \lambda -2 c \right ) \sqrt {-\frac {b}{a}}}{a}, \frac {\sqrt {-a b \left (4 \sqrt {-a b}\, a^{2} d -4 \sqrt {-a b}\, a b c -4 a^{3} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{2 a^{2} b}, \frac {\sqrt {4 a^{2} b \left (a d -b c \right ) \sqrt {-a b}+4 a^{4} b^{2}-a^{3} b \,d^{2}+2 d \,b^{2} c \,a^{2}-b^{3} c^{2} a}}{2 a^{2} b}, 0, \frac {\lambda d}{a}-\frac {b c \lambda }{a^{2}}+\frac {1}{2}-\frac {d^{2}}{8 a b}-\frac {c d}{4 a^{2}}+\frac {3 b \,c^{2}}{8 a^{3}}, \frac {a x}{2 \sqrt {-a b}}+\frac {1}{2}\right )\right ) \]

Problem 10993

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

Problem 10994

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

Problem 10995

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

Problem 10996

ODE	\[ \boxed {2 x \left (x -1\right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (a x +b \right ) y=0} \]
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 10997

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

Problem 10998

ODE	\[ \boxed {\left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (d x +k \right ) y^{\prime }+\left (-2 a +d \right ) y=0} \]
program solution	\[ y = \left (a \,x^{2}+b x +c \right )^{\frac {2 a -d}{2 a}} {\mathrm e}^{-\frac {2 \left (a k -\frac {b d}{2}\right ) \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a}} \left (c_{1} \left (\int \left (a \,x^{2}+b x +c \right )^{\frac {-4 a +d}{2 a}} {\mathrm e}^{\frac {\left (2 a k -b d \right ) \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{a \sqrt {4 a c -b^{2}}}}d x \right )+c_{2} \right ) \] Verified OK.
Maple solution	\[ \text {Expression too large to display} \]

Problem 10999

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

Problem 11000

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