Internal problem ID [8097]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 621.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+x \left (-2 x^{2}+1\right ) y^{\prime }-4 \left (2 x^{2}+1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 51
dsolve(x^2*diff(y(x),x$2)+x*(1-2*x^2)*diff(y(x),x)-4*(1+2*x^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{2} {\mathrm e}^{x^{2}}+\frac {c_{2} {\mathrm e}^{x^{2}} \left (\operatorname {expIntegral}_{1}\left (x^{2}\right ) x^{4}-{\mathrm e}^{-x^{2}} x^{2}+{\mathrm e}^{-x^{2}}\right )}{4 x^{2}} \]
✓ Solution by Mathematica
Time used: 0.067 (sec). Leaf size: 46
DSolve[x^2*y''[x]+x*(1-2*x^2)*y'[x]-4*(1+2*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {c_2 \left (e^{x^2} x^4 \operatorname {ExpIntegralEi}\left (-x^2\right )+x^2-1\right )}{4 x^2}+c_1 e^{x^2} x^2 \]