Internal problem ID [8949]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1370.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {2 x y^{\prime }}{x^{2}-1}-\frac {a^{2} y}{\left (x^{2}-1\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 19
dsolve(diff(diff(y(x),x),x) = -2*x/(x^2-1)*diff(y(x),x)+a^2/(x^2-1)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sinh \left (a \arctanh \relax (x )\right )+c_{2} \cosh \left (a \arctanh \relax (x )\right ) \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 25
DSolve[y''[x] == (a^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 \cosh \left (a \tanh ^{-1}(x)\right )-i c_2 \sinh \left (a \tanh ^{-1}(x)\right ) \\ \end{align*}