Internal problem ID [7441]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 722.
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 }+x y^{\prime }+\frac {\left (-9 a^{2}+4 x^{2}\right ) y}{4 a^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.171 (sec). Leaf size: 45
dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(4*x^2-9*a^2)/(4*a^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{\frac {i x}{a}} \left (i x -a \right )}{x^{\frac {3}{2}}}+\frac {c_{2} {\mathrm e}^{-\frac {i x}{a}} \left (i x +a \right )}{x^{\frac {3}{2}}} \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 62
DSolve[x^2*y''[x]+x*y'[x]+(4*x^2-9*a^2)/(4*a^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {\frac {2}{\pi }} \left ((a c_2+c_1 x) \cos \left (\frac {x}{a}\right )+(c_2 x-a c_1) \sin \left (\frac {x}{a}\right )\right )}{x \sqrt {\frac {x}{a}}} \\ \end{align*}