34.6.7 problem 7

Internal problem ID [6081]
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
Date solved : Monday, January 27, 2025 at 01:35:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} u^{\prime \prime }+\frac {4 u^{\prime }}{x}+a^{2} u&=0 \end{align*}

✓ 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)
 

\[ u = \frac {\left (x c_1 a +c_2 \right ) \cos \left (a x \right )+\sin \left (a x \right ) \left (c_2 a x -c_1 \right )}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.073 (sec). Leaf size: 57

DSolve[D[u[x],{x,2}]+4/x*D[u[x],x]+a^2*u[x]==0,u[x],x,IncludeSingularSolutions -> True]
 

\[ u(x)\to -\frac {\sqrt {\frac {2}{\pi }} ((a c_1 x+c_2) \cos (a x)+(a c_2 x-c_1) \sin (a x))}{x^{3/2} (a x)^{3/2}} \]