Internal problem ID [6919]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 187.
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 \left (-2 x^{2}+1\right ) y^{\prime }-4 \left (2 x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 38
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)
\[ y \relax (x ) = c_{1} x^{2} {\mathrm e}^{x^{2}}+\frac {c_{2} \left (-x^{4} {\mathrm e}^{x^{2}} \expIntegral \left (1, x^{2}\right )+x^{2}-1\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.067 (sec). Leaf size: 44
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]
\begin{align*} y(x)\to \frac {e^{x^2} x^4 \left (c_2 \text {Ei}\left (-x^2\right )+4 c_1\right )+c_2 \left (x^2-1\right )}{4 x^2} \\ \end{align*}