Internal problem ID [8813]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1235.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {\left (x^{2}-1\right ) y^{\prime \prime }+y^{\prime } x +a y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 43
dsolve((x^2-1)*diff(diff(y(x),x),x)+x*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (x +\sqrt {x^{2}-1}\right )^{i \sqrt {a}}+c_{2} \left (x +\sqrt {x^{2}-1}\right )^{-i \sqrt {a}} \]
✓ Solution by Mathematica
Time used: 0.026 (sec). Leaf size: 50
DSolve[a*y[x] + x*y'[x] + (-1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \cos \left (\sqrt {a} \text {arctanh}\left (\frac {x}{\sqrt {x^2-1}}\right )\right )+c_2 \sin \left (\sqrt {a} \text {arctanh}\left (\frac {x}{\sqrt {x^2-1}}\right )\right ) \\ \end{align*}