Internal problem ID [7205]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 484.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (2 x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+4\right ) y^{\prime }+2 \left (-x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 43
dsolve(x^2*(1+2*x^2)*diff(y(x),x$2)+x*(4+2*x^2)*diff(y(x),x)+2*(1-x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1}}{x}+\frac {c_{2} \left (3 \arcsinh \left (x \sqrt {2}\right ) x +\sqrt {2}\, \sqrt {2 x^{2}+1}\, \left (x^{2}-1\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.058 (sec). Leaf size: 67
DSolve[x^2*(1+2*x^2)*y''[x]+x*(4+2*x^2)*y'[x]+2*(1-x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {-2 c_2 \sqrt {2 x^2+1}+2 x \left (c_2 x \sqrt {2 x^2+1}+c_1\right )+3 \sqrt {2} c_2 x \sinh ^{-1}\left (\sqrt {2} x\right )}{2 x^2} \\ \end{align*}