Internal problem ID [6882]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 150.
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 (-x^{2}+1\right ) y^{\prime }+\left (x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.006 (sec). Leaf size: 30
dsolve(x^2*diff(y(x),x$2)-x*(1-x^2)*diff(y(x),x)+(1+x^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x \,{\mathrm e}^{-\frac {x^{2}}{2}}+c_{2} x \,{\mathrm e}^{-\frac {x^{2}}{2}} \expIntegral \left (1, -\frac {x^{2}}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.008 (sec). Leaf size: 35
DSolve[x^2*y''[x]-x*(1-x^2)*y'[x]+(1+x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} e^{-\frac {x^2}{2}} x \left (c_1 \operatorname {Ei}\left (\frac {x^2}{2}\right )+2 c_2\right ) \\ \end{align*}