Internal problem ID [10983]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-5 Equation of form
\((a x^2+b x+c) y''+f(x)y'+g(x)y=0\)
Problem number: 149.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }+n \left (-1+n \right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 57
dsolve((1-x^2)*diff(y(x),x$2)+n*(n-1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (-x^{2}+1\right ) \operatorname {hypergeom}\left (\left [-\frac {n}{2}+1, \frac {n}{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right )+c_{2} \left (-x^{3}+x \right ) \operatorname {hypergeom}\left (\left [1+\frac {n}{2}, \frac {3}{2}-\frac {n}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.149 (sec). Leaf size: 56
DSolve[(1-x^2)*y''[x]+n*(n-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to i c_2 x \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {n}{2},\frac {n}{2},\frac {3}{2},x^2\right )+c_1 \operatorname {Hypergeometric2F1}\left (\frac {n-1}{2},-\frac {n}{2},\frac {1}{2},x^2\right ) \]