Internal problem ID [8770]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1192.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (x^{2}-1\right ) y^{\prime }-y=0} \]
✓ Solution by Maple
Time used: 0.25 (sec). Leaf size: 53
dsolve(x^2*diff(diff(y(x),x),x)+(x^2-1)*diff(y(x),x)-y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sqrt {x}\, {\mathrm e}^{-x} \operatorname {HeunD}\left (4, 3, -8, 5, \frac {x -1}{x +1}\right )+c_{2} \sqrt {x}\, {\mathrm e}^{-\frac {1}{x}} \operatorname {HeunD}\left (-4, 3, -8, 5, \frac {x -1}{x +1}\right ) \]
✓ Solution by Mathematica
Time used: 0.061 (sec). Leaf size: 35
DSolve[-y[x] + (-1 + x^2)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-x} \left (c_2 \int _1^xe^{K[1]-\frac {1}{K[1]}}dK[1]+c_1\right ) \\ \end{align*}