Internal problem ID [6940]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 209.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Lienard]
Solve \begin {gather*} \boxed {t y^{\prime \prime }-\left (t^{2}+2\right ) y^{\prime }+y t=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 40
dsolve(t*diff(y(t),t$2)-(t^2+2)*diff(y(t),t)+t*y(t)=0,y(t), singsol=all)
\[ y \relax (t ) = c_{1} {\mathrm e}^{\frac {t^{2}}{2}}+c_{2} \left (\sqrt {\pi }\, t \sqrt {2}-\erf \left (\frac {t \sqrt {2}}{2}\right ) {\mathrm e}^{\frac {t^{2}}{2}} \pi \right ) \]
✓ Solution by Mathematica
Time used: 0.081 (sec). Leaf size: 47
DSolve[t*y''[t]-(t^2+2)*y'[t]+t*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{2} e^{\frac {t^2}{2}} \left (\sqrt {2 \pi } c_2 \text {Erf}\left (\frac {t}{\sqrt {2}}\right )+2 c_1\right )-c_2 t \\ \end{align*}