2.17.111 Problems 11001 to 11100

Problem 11001

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

Problem 11002

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

Problem 11003

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

Problem 11004

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

Problem 11005

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

Problem 11006

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

Problem 11007

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

Problem 11008

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

Problem 11009

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

Problem 11010

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

Problem 11011

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

Problem 11012

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

Problem 11013

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

Problem 11014

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

Problem 11015

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

Problem 11016

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

Problem 11017

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

Problem 11018

ODE	\[ \boxed {\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }+\left (\beta -2 b \right ) y=0} \]
program solution
Maple solution	\[ \text {Expression too large to display} \]

Problem 11019

ODE	\[ \boxed {\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }-\left (x \alpha +2 b -\beta \right ) y=0} \]
program solution
Maple solution	\[ \text {Expression too large to display} \]

Problem 11020

ODE	\[ \boxed {\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (-2 a \,x^{2}-\left (1+b \right ) x +k \right ) y^{\prime }+2 \left (a x +1\right ) y=0} \]
program solution
Maple solution	\[ \text {Expression too large to display} \]

Problem 11021

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

Problem 11022

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

Problem 11023

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

Problem 11024

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

Problem 11025

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

Problem 11026

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

Problem 11027

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

Problem 11028

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

Problem 11029

ODE	\[ \boxed {2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\lambda y=0} \]
program solution	\[ \text {Expression too large to display} \] Warning, solution could not be verified
Maple solution	\[ y \left (x \right ) = c_{1} {\mathrm e}^{\frac {i \sqrt {2}\, \sqrt {\lambda }\, \left (\int \frac {1}{\sqrt {a \,x^{3}+x^{2} b +c x +d}}d x \right )}{2}}+c_{2} {\mathrm e}^{-\frac {i \sqrt {2}\, \sqrt {\lambda }\, \left (\int \frac {1}{\sqrt {a \,x^{3}+x^{2} b +c x +d}}d x \right )}{2}} \]

Problem 11030

ODE	\[ \boxed {2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+3 \left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\left (6 a x +2 b +\lambda \right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{\frac {\sqrt {2}\, \sqrt {-\frac {\lambda }{a}}\, \left (\int \frac {1}{\sqrt {\frac {a \,x^{3}+x^{2} b +c x +d}{a}}}d x \right )}{2}}+c_{2} {\mathrm e}^{-\frac {\sqrt {2}\, \sqrt {-\frac {\lambda }{a}}\, \left (\int \frac {1}{\sqrt {\frac {a \,x^{3}+x^{2} b +c x +d}{a}}}d x \right )}{2}}}{\sqrt {a \,x^{3}+x^{2} b +c x +d}} \]

Problem 11031

ODE	\[ \boxed {\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\left (\alpha \gamma +\beta \right ) x +\beta \lambda \right ) y^{\prime }-\left (x \alpha +\beta \right ) y=0} \]
program solution
Maple solution	\[ \text {No solution found} \]

Problem 11032

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

Problem 11033

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

Problem 11034

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

Problem 11035

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

Problem 11036

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

Problem 11037

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

Problem 11038

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

Problem 11039

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

Problem 11040

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

Problem 11041

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

Problem 11042

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

Problem 11043

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

Problem 11044

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

Problem 11045

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

Problem 11046

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

Problem 11047

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

Problem 11048

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

Problem 11049

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

Problem 11050

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

Problem 11051

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

Problem 11052

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

Problem 11053

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

Problem 11054

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

Problem 11055

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

Problem 11056

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

Problem 11057

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

Problem 11058

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

Problem 11059

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

Problem 11060

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

Problem 11061

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

Problem 11062

ODE	\[ \boxed {x^{6} y^{\prime \prime }-x^{5} y^{\prime }+a y=0} \]
program solution	\[ y = \frac {2 c_{1} x \sin \left (\frac {\sqrt {a}}{2 x^{2}}\right )}{\sqrt {\pi }\, \sqrt {\frac {\sqrt {a}}{x^{2}}}}-\frac {2 c_{2} x \cos \left (\frac {\sqrt {a}}{2 x^{2}}\right )}{\sqrt {\pi }\, \sqrt {\frac {\sqrt {a}}{x^{2}}}} \] Verified OK.
Maple solution	\[ y \left (x \right ) = x^{2} \left (c_{1} \sinh \left (\frac {\sqrt {-a}}{2 x^{2}}\right )+c_{2} \cosh \left (\frac {\sqrt {-a}}{2 x^{2}}\right )\right ) \]

Problem 11063

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

Problem 11064

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

Problem 11065

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

Problem 11066

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

Problem 11067

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

Problem 11068

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

Problem 11069

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

Problem 11070

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

Problem 11071

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

Problem 11072

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

Problem 11073

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

Problem 11074

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

Problem 11075

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

Problem 11076

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

Problem 11077

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

Problem 11078

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

Problem 11079

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

Problem 11080

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

Problem 11081

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

Problem 11082

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

Problem 11083

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

Problem 11084

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

Problem 11085

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

Problem 11086

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

Problem 11087

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

Problem 11088

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

Problem 11089

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

Problem 11090

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

Problem 11091

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

Problem 11092

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

Problem 11093

ODE	\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{4 \lambda x}+b \,{\mathrm e}^{3 \lambda x}+{\mathrm e}^{2 \lambda x} c -\frac {\lambda ^{2}}{4}\right ) y=0} \]
program solution
Maple solution	\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [\frac {4 \lambda \,a^{\frac {3}{2}}+4 i a c -i b^{2}}{16 \lambda \,a^{\frac {3}{2}}}\right ], \left [\frac {1}{2}\right ], \frac {i \left (2 \,{\mathrm e}^{x \lambda } a +b \right )^{2}}{4 \lambda \,a^{\frac {3}{2}}}\right ) {\mathrm e}^{-\frac {i {\mathrm e}^{2 x \lambda } a +\lambda ^{2} x \sqrt {a}+i b \,{\mathrm e}^{x \lambda }}{2 \lambda \sqrt {a}}}+c_{2} \operatorname {hypergeom}\left (\left [\frac {12 \lambda \,a^{\frac {3}{2}}+4 i a c -i b^{2}}{16 \lambda \,a^{\frac {3}{2}}}\right ], \left [\frac {3}{2}\right ], \frac {i \left (2 \,{\mathrm e}^{x \lambda } a +b \right )^{2}}{4 \lambda \,a^{\frac {3}{2}}}\right ) \left (2 a \,{\mathrm e}^{-\frac {i {\mathrm e}^{2 x \lambda } a -\lambda ^{2} x \sqrt {a}+i b \,{\mathrm e}^{x \lambda }}{2 \lambda \sqrt {a}}}+b \,{\mathrm e}^{-\frac {i {\mathrm e}^{2 x \lambda } a +\lambda ^{2} x \sqrt {a}+i b \,{\mathrm e}^{x \lambda }}{2 \lambda \sqrt {a}}}\right ) \]

Problem 11094

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

Problem 11095

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

Problem 11096

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

Problem 11097

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

Problem 11098

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

Problem 11099

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

Problem 11100

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