Internal problem ID [4737]
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]]
\[ \boxed {u^{\prime \prime }+\frac {4 u^{\prime }}{x}+a^{2} u=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 33
dsolve(diff(u(x),x$2)+4/x*diff(u(x),x)+a^2*u(x)=0,u(x), singsol=all)
\[ u \left (x \right ) = \frac {\left (a c_{1} x +c_{2} \right ) \cos \left (a x \right )+\sin \left (a x \right ) \left (a c_{2} x -c_{1} \right )}{x^{3}} \]
✓ Solution by Mathematica
Time used: 0.094 (sec). Leaf size: 57
DSolve[u''[x]+4/x*u'[x]+a^2*u[x]==0,u[x],x,IncludeSingularSolutions -> True]
\[ u(x)\to -\frac {\sqrt {\frac {2}{\pi }} ((a c_1 x+c_2) \cos (a x)+(a c_2 x-c_1) \sin (a x))}{x^{3/2} (a x)^{3/2}} \]