Internal problem ID [7433]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 714.
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: 28
dsolve(diff(u(x),x$2)-2/x*diff(u(x),x)-a^2*u(x)=0,u(x), singsol=all)
\[ u \relax (x ) = c_{1} {\mathrm e}^{a x} \left (a x -1\right )+c_{2} {\mathrm e}^{-a x} \left (a x +1\right ) \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 68
DSolve[u''[x]-2/x*u'[x]-a^2*u[x]==0,u[x],x,IncludeSingularSolutions -> True]
\begin{align*} u(x)\to \frac {\sqrt {\frac {2}{\pi }} \sqrt {x} ((i a c_2 x+c_1) \sinh (a x)-(a c_1 x+i c_2) \cosh (a x))}{a \sqrt {-i a x}} \\ \end{align*}