Internal problem ID [4234]
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: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {25}{4}\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.09 (sec). Leaf size: 45
dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-25/4)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{i x} \left (x^{2}+3 i x -3\right )}{x^{\frac {5}{2}}}+\frac {c_{2} {\mathrm e}^{-i x} \left (-x^{2}+3 i x +3\right )}{x^{\frac {5}{2}}} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 57
DSolve[x^2*y''[x]+x*y'[x]+(x^2-25/4)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {\sqrt {\frac {2}{\pi }} \left (\left (3 c_1 x-c_2 \left (x^2-3\right )\right ) \cos (x)+\left (c_1 \left (x^2-3\right )+3 c_2 x\right ) \sin (x)\right )}{x^{5/2}} \\ \end{align*}