Internal problem ID [8950]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1371.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {2 x y^{\prime }}{x^{2}-1}+\frac {\left (-a^{2}-\lambda \left (x^{2}-1\right )\right ) y}{\left (x^{2}-1\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 37
dsolve(diff(diff(y(x),x),x) = -2*x/(x^2-1)*diff(y(x),x)-(-a^2-lambda*(x^2-1))/(x^2-1)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \LegendreP \left (\frac {\sqrt {1+4 \lambda }}{2}-\frac {1}{2}, a , x\right )+c_{2} \LegendreQ \left (\frac {\sqrt {1+4 \lambda }}{2}-\frac {1}{2}, a , x\right ) \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 48
DSolve[y''[x] == -(((-a^2 - \[Lambda]*(-1 + x^2))*y[x])/(-1 + x^2)^2) - (2*x*y'[x])/(-1 + x^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 P_{\frac {1}{2} \left (\sqrt {4 \lambda +1}-1\right )}^a(x)+c_2 Q_{\frac {1}{2} \left (\sqrt {4 \lambda +1}-1\right )}^a(x) \\ \end{align*}