Internal problem ID [7435]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 716.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {u^{\prime \prime }+\frac {2 u^{\prime }}{x}+a^{2} u=0} \end {gather*}
✓ Solution by Maple
Time used: 0.005 (sec). Leaf size: 23
dsolve(diff(u(x),x$2)+2/x*diff(u(x),x)+a^2*u(x)=0,u(x), singsol=all)
\[ u \relax (x ) = \frac {c_{1} \sin \left (a x \right )}{x}+\frac {c_{2} \cos \left (a x \right )}{x} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 42
DSolve[u''[x]+2/x*u'[x]+a^2*u[x]==0,u[x],x,IncludeSingularSolutions -> True]
\begin{align*} u(x)\to \frac {e^{-i a x} \left (2 c_1-\frac {i c_2 e^{2 i a x}}{a}\right )}{2 x} \\ \end{align*}