Internal problem ID [8731]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1151.
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 (a \,x^{2}+2\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.014 (sec). Leaf size: 43
dsolve(x^2*diff(diff(y(x),x),x)-(a*x^2+2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{x \sqrt {a}} \left (-a x +\sqrt {a}\right )}{x}+\frac {c_{2} {\mathrm e}^{-x \sqrt {a}} \left (a x +\sqrt {a}\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 88
DSolve[(-2 - a*x^2)*y[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i \sqrt {\frac {2}{\pi }} \sqrt {x} \left (\left (i \sqrt {a} c_2 x+c_1\right ) \sinh \left (\sqrt {a} x\right )-\left (\sqrt {a} c_1 x+i c_2\right ) \cosh \left (\sqrt {a} x\right )\right )}{\left (-i \sqrt {a} x\right )^{3/2}} \\ \end{align*}