Internal problem ID [7007]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 276.
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: 0.007 (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)
\[ u \relax (x ) = \frac {c_{1} \sinh \left (a x \right )}{x}+\frac {c_{2} \cosh \left (a x \right )}{x} \]
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 35
DSolve[u''[x]+2/x*u'[x]-a^2*u[x]==0,u[x],x,IncludeSingularSolutions -> True]
\begin{align*} u(x)\to \frac {2 a c_1 e^{-a x}+c_2 e^{a x}}{2 a x} \\ \end{align*}