Internal problem ID [10990]
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: 156.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer]
\[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+2 \left (1+n \right ) x y^{\prime }-\left (\nu +n +1\right ) \left (\nu -n \right ) y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 35
dsolve((x^2-1)*diff(y(x),x$2)+2*(n+1)*x*diff(y(x),x)-(nu+n+1)*(nu-n)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (x^{2}-1\right )^{-\frac {n}{2}} \operatorname {LegendreP}\left (\nu , n , x\right )+c_{2} \left (x^{2}-1\right )^{-\frac {n}{2}} \operatorname {LegendreQ}\left (\nu , n , x\right ) \]
✓ Solution by Mathematica
Time used: 0.069 (sec). Leaf size: 32
DSolve[(x^2-1)*y''[x]+2*(n+1)*x*y'[x]-(\[Nu]+n+1)*(\[Nu]-n)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (x^2-1\right )^{-n/2} (c_1 P_{\nu }^n(x)+c_2 Q_{\nu }^n(x)) \]