34.6.8 problem 8

Internal problem ID [6082]
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 : 8
Date solved : Monday, January 27, 2025 at 01:35:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-a^{2} y&=\frac {6 y}{x^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 48

dsolve(diff(y(x),x$2)-a^2*y(x)=6*y(x)/x^2,y(x), singsol=all)
 

\[ y = \frac {c_2 \,{\mathrm e}^{-a x} \left (a^{2} x^{2}+3 a x +3\right )+c_1 \,{\mathrm e}^{a x} \left (a^{2} x^{2}-3 a x +3\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.126 (sec). Leaf size: 90

DSolve[D[y[x],{x,2}]-a^2*y[x]==6*y[x]/x^2,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\sqrt {\frac {2}{\pi }} \left (\left (a^2 c_2 x^2-3 i a c_1 x+3 c_2\right ) \cosh (a x)+i \left (c_1 \left (a^2 x^2+3\right )+3 i a c_2 x\right ) \sinh (a x)\right )}{a^2 x^{3/2} \sqrt {-i a x}} \]