Internal problem ID [8633]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1053.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+2 a x y^{\prime }+y a^{2} x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 35
dsolve(diff(diff(y(x),x),x)+2*a*x*diff(y(x),x)+a^2*x^2*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-\frac {x \left (a x -2 \sqrt {a}\right )}{2}}+c_{2} {\mathrm e}^{-\frac {x \left (a x +2 \sqrt {a}\right )}{2}} \]
✓ Solution by Mathematica
Time used: 0.02 (sec). Leaf size: 56
DSolve[a^2*x^2*y[x] + 2*a*x*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-\frac {a x^2}{2}-\sqrt {a} x} \left (c_2 e^{2 \sqrt {a} x}+2 \sqrt {a} c_1\right )}{2 \sqrt {a}} \\ \end{align*}