Internal problem ID [8759]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1179.
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 }-2 x y^{\prime }+\left (a^{2} x^{2}+2\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 19
dsolve(x^2*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+(a^2*x^2+2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x \sin \left (a x \right )+c_{2} x \cos \left (a x \right ) \]
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 38
DSolve[(2 + a^2*x^2)*y[x] - 2*x*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 x e^{-i a x}-\frac {i c_2 x e^{i a x}}{2 a} \\ \end{align*}