Internal problem ID [4228]
Book: A treatise on ordinary and partial differential equations by William Woolsey Johnson.
1913
Section: Chapter IX, Special forms of differential equations. Examples XVII. page 247
Problem number: 6.
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 {4 u^{\prime }}{x}-a^{2} u=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 34
dsolve(diff(u(x),x$2)+4/x*diff(u(x),x)-a^2*u(x)=0,u(x), singsol=all)
\[ u \relax (x ) = \frac {c_{1} {\mathrm e}^{a x} \left (a x -1\right )}{x^{3}}+\frac {c_{2} {\mathrm e}^{-a x} \left (a x +1\right )}{x^{3}} \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 68
DSolve[u''[x]+4/x*u'[x]-a^2*u[x]==0,u[x],x,IncludeSingularSolutions -> True]
\begin{align*} u(x)\to \frac {\sqrt {\frac {2}{\pi }} ((i a c_2 x+c_1) \sinh (a x)-(a c_1 x+i c_2) \cosh (a x))}{a x^{5/2} \sqrt {-i a x}} \\ \end{align*}