Internal problem ID [6857]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 125.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }-2 x \left (2 x^{2}+1\right ) y^{\prime }+\left (-2 x^{2}+2\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.097 (sec). Leaf size: 51
dsolve(x^2*(2-x^2)*diff(y(x),x$2)-2*x*(1+2*x^2)*diff(y(x),x)+(2-2*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} x}{\left (x^{2}-2\right )^{\frac {3}{2}}}+\frac {c_{2} x \left (2 \arctan \left (\frac {\sqrt {2}}{\sqrt {x^{2}-2}}\right )+\sqrt {2}\, \sqrt {x^{2}-2}\right )}{\left (x^{2}-2\right )^{\frac {3}{2}}} \]
✓ Solution by Mathematica
Time used: 0.069 (sec). Leaf size: 58
DSolve[x^2*(2-x^2)*y''[x]-2*x*(1+2*x^2)*y'[x]+(2-2*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x \left (c_2 \sqrt {2-x^2}-\sqrt {2} c_2 \tanh ^{-1}\left (\sqrt {1-\frac {x^2}{2}}\right )+c_1\right )}{\left (2-x^2\right )^{3/2}} \\ \end{align*}