34.6.6 problem 6

Internal problem ID [6080]
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
Date solved : Monday, January 27, 2025 at 01:35:05 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: 0.006 (sec). Leaf size: 32

dsolve(diff(u(x),x$2)+4/x*diff(u(x),x)-a^2*u(x)=0,u(x), singsol=all)
 

\[ u = \frac {c_1 \,{\mathrm e}^{a x} \left (a x -1\right )+c_2 \,{\mathrm e}^{-a x} \left (a x +1\right )}{x^{3}} \]

✓ Solution by Mathematica

Time used: 0.071 (sec). Leaf size: 68

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

\[ u(x)\to \frac {\sqrt {\frac {2}{\pi }} ((i a c_2 x+c_1) \sinh (a x)-(a c_1 x+i c_2) \cosh (a x))}{a x^{5/2} \sqrt {-i a x}} \]