Internal problem ID [6746]
Book: Collection of Kovacic problems
Section: section 1
Problem number: 12.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {t y^{\prime \prime }+\left (t^{2}-1\right ) y^{\prime }+t^{2} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 57
dsolve(t*diff(y(t),t$2)+ (t^2-1)*diff(y(t),t)+t^2*y(t) = 0,y(t), singsol=all)
\[ y \relax (t ) = c_{1} {\mathrm e}^{-\frac {t \left (t -2\right )}{2}} \left (t -1\right )+c_{2} \left (\hypergeom \left (\left [\frac {1}{2}\right ], \left [\frac {3}{2}\right ], \frac {\left (t -2\right )^{2}}{2}\right ) \left (t -2\right )-\hypergeom \left (\left [-\frac {1}{2}\right ], \left [\frac {1}{2}\right ], \frac {\left (t -2\right )^{2}}{2}\right )\right ) {\mathrm e}^{-\frac {t \left (t -2\right )}{2}} \]
✓ Solution by Mathematica
Time used: 0.235 (sec). Leaf size: 69
DSolve[t*y''[t]+(t^2-1)*y'[t]+t^2*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \frac {1}{2} e^{-\frac {t^2}{2}+t-2} \left ((t-1) \left (\sqrt {2 \pi } c_2 \text {Erfi}\left (\frac {t-2}{\sqrt {2}}\right )+2 e^2 c_1\right )-2 c_2 e^{\frac {1}{2} (t-2)^2}\right ) \\ \end{align*}