Internal problem ID [8789]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1209.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+\left (x^{2}+2\right ) x y^{\prime }+\left (x^{2}-2\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 48
dsolve(x^2*diff(diff(y(x),x),x)+(x^2+2)*x*diff(y(x),x)+(x^2-2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{-\frac {x^{2}}{2}}}{x^{2}}+\frac {c_{2} \left (i \sqrt {2}\, \sqrt {\pi }\, {\mathrm e}^{-\frac {x^{2}}{2}} \erf \left (\frac {i \sqrt {2}\, x}{2}\right )+2 x \right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.026 (sec). Leaf size: 42
DSolve[(-2 + x^2)*y[x] + x*(2 + x^2)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 \left (x-\sqrt {2} F\left (\frac {x}{\sqrt {2}}\right )\right )+c_2 e^{-\frac {x^2}{2}}}{x^2} \\ \end{align*}