Internal problem ID [10987]
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: 153.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-2 y^{\prime } x +n \left (1+n \right ) y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 15
dsolve((1-x^2)*diff(y(x),x$2)-2*x*diff(y(x),x)+n*(n+1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {LegendreP}\left (n , x\right )+c_{2} \operatorname {LegendreQ}\left (n , x\right ) \]
✓ Solution by Mathematica
Time used: 0.042 (sec). Leaf size: 18
DSolve[(1-x^2)*y''[x]-2*x*y'[x]+n*(n+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \operatorname {LegendreP}(n,x)+c_2 \operatorname {LegendreQ}(n,x) \]