34.6.5 problem 5

Internal problem ID [6079]
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 : 5
Date solved : Monday, January 27, 2025 at 01:35:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 21

dsolve(diff(u(x),x$2)+2/x*diff(u(x),x)+a^2*u(x)=0,u(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.045 (sec). Leaf size: 42

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

\[ u(x)\to \frac {e^{-i a x} \left (2 c_1-\frac {i c_2 e^{2 i a x}}{a}\right )}{2 x} \]