Internal problem ID [9482]
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]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (a^{2} x^{2}-6\right ) y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 53
dsolve(x^2*diff(diff(y(x),x),x)+(a^2*x^2-6)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (a^{2} c_{1} x^{2}+3 a c_{2} x -3 c_{1} \right ) \cos \left (a x \right )+\sin \left (a x \right ) \left (a^{2} c_{2} x^{2}-3 a c_{1} x -3 c_{2} \right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.165 (sec). Leaf size: 79
DSolve[(-6 + a^2*x^2)*y[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {\sqrt {\frac {2}{\pi }} \sqrt {x} \left (\left (-a^2 c_2 x^2+3 a c_1 x+3 c_2\right ) \cos (a x)+\left (c_1 \left (a^2 x^2-3\right )+3 a c_2 x\right ) \sin (a x)\right )}{(a x)^{5/2}} \]