Internal problem ID [4230]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-a^{2} y-\frac {6 y}{x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 50
dsolve(diff(y(x),x$2)-a^2*y(x)=6*y(x)/x^2,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{a x} \left (a^{2} x^{2}-3 a x +3\right )}{x^{2}}+\frac {c_{2} {\mathrm e}^{-a x} \left (a^{2} x^{2}+3 a x +3\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 88
DSolve[y''[x]-a^2*y[x]==6*y[x]/x^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {\frac {2}{\pi }} \left (i \left (c_1 \left (a^2 x^2+3\right )+3 i a c_2 x\right ) \sinh (a x)+(a x (a c_2 x-3 i c_1)+3 c_2) \cosh (a x)\right )}{a^2 x^{3/2} \sqrt {-i a x}} \\ \end{align*}