Internal problem ID [10989]
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: 155.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-3 y^{\prime } x +n y \left (2+n \right )=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 62
dsolve((1-x^2)*diff(y(x),x$2)-3*x*diff(y(x),x)+n*(n+2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} \left (x +\sqrt {x^{2}-1}\right )^{-1-n}}{\sqrt {x^{2}-1}}+\frac {c_{2} \left (x +\sqrt {x^{2}-1}\right )^{n}}{\left (\sqrt {x^{2}-1}-x \right ) \sqrt {x^{2}-1}} \]
✓ Solution by Mathematica
Time used: 0.061 (sec). Leaf size: 42
DSolve[(1-x^2)*y''[x]-3*x*y'[x]+n*(n+2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {c_1 P_{n+\frac {1}{2}}^{\frac {1}{2}}(x)+c_2 Q_{n+\frac {1}{2}}^{\frac {1}{2}}(x)}{\sqrt [4]{x^2-1}} \]