Internal problem ID [7571]
Book: Collection of Kovacic problems
Section: section 3. Problems from Kovacic related papers
Problem number: Kovacic 1985 paper. page 14. section 3.2, example 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\left (\frac {6}{x^{2}}-1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 47
dsolve(diff(y(x),x$2)= ( (4*(5/2)^2-1)/(4*x^2)-1)*y(x),y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (\cos \relax (x ) x^{2}-3 \sin \relax (x ) x -3 \cos \relax (x )\right )}{x^{2}}+\frac {c_{2} \left (x^{2} \sin \relax (x )+3 x \cos \relax (x )-3 \sin \relax (x )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 55
DSolve[y''[x]== ( (4*(5/2)^2-1)/(4*x^2)-1)*y[x],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^2} \\ \end{align*}