Internal problem ID [8732]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1152.
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^{2} x^{2}-6\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 61
dsolve(x^2*diff(diff(y(x),x),x)+(a^2*x^2-6)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (\left (a^{2} x^{2}-3\right ) \cos \left (a x \right )-3 \sin \left (a x \right ) a x \right )}{x^{2}}+\frac {c_{2} \left (3 \cos \left (a x \right ) a x +\left (a^{2} x^{2}-3\right ) \sin \left (a x \right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 77
DSolve[(-6 + a^2*x^2)*y[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {\frac {2}{\pi }} \sqrt {x} \left (\left (c_1 \left (a^2 x^2-3\right )+3 a c_2 x\right ) \sin (a x)+(a x (3 c_1-a c_2 x)+3 c_2) \cos (a x)\right )}{(a x)^{5/2}} \\ \end{align*}