Internal problem ID [4229]
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: 7.
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: 0.036 (sec). Leaf size: 41
dsolve(diff(u(x),x$2)+4/x*diff(u(x),x)+a^2*u(x)=0,u(x), singsol=all)
\[ u \relax (x ) = \frac {c_{1} \left (\cos \left (a x \right ) a x -\sin \left (a x \right )\right )}{x^{3}}+\frac {c_{2} \left (\cos \left (a x \right )+\sin \left (a x \right ) a x \right )}{x^{3}} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 57
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 }} ((a c_1 x+c_2) \cos (a x)+(a c_2 x-c_1) \sin (a x))}{x^{3/2} (a x)^{3/2}} \\ \end{align*}