Internal problem ID [7340]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 620.
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 }-\left (-10 x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.102 (sec). Leaf size: 37
dsolve(x^2*diff(y(x),x$2)+x*(1+2*x^2)*diff(y(x),x)-(1-10*x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{-x^{2}} x \hypergeom \left (\left [-1\right ], \relax [2], x^{2}\right )+c_{2} {\mathrm e}^{-x^{2}} x \KummerU \left (-1, 2, x^{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.066 (sec). Leaf size: 49
DSolve[x^2*y''[x]+x*(1+2*x^2)*y'[x]-(1-10*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{4} e^{-x^2} x \left (x^2-2\right ) \left (c_2 \operatorname {Ei}\left (x^2\right )+4 c_1\right )-\frac {c_2 \left (x^2-1\right )}{4 x} \\ \end{align*}