Internal problem ID [7502]
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]]
\[ \boxed {t y^{\prime \prime }+\left (t^{2}-1\right ) y^{\prime }+t^{2} y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 50
dsolve(t*diff(y(t),t$2)+ (t^2-1)*diff(y(t),t)+t^2*y(t) = 0,y(t), singsol=all)
\[ y \left (t \right ) = c_{1} {\mathrm e}^{t -\frac {1}{2} t^{2}} \left (t -1\right )+c_{2} {\mathrm e}^{t -\frac {1}{2} t^{2}} \left (t -1\right ) \left (\int \frac {t \,{\mathrm e}^{\frac {1}{2} t^{2}-2 t}}{\left (t -1\right )^{2}}d t \right ) \]
✓ Solution by Mathematica
Time used: 0.434 (sec). Leaf size: 70
DSolve[t*y''[t]+(t^2-1)*y'[t]+t^2*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \frac {1}{2} e^{-\frac {t^2}{2}+t-2} \left (\sqrt {2 \pi } c_2 (t-1) \text {erfi}\left (\frac {t-2}{\sqrt {2}}\right )+2 e^2 c_1 (t-1)-2 c_2 e^{\frac {1}{2} (t-2)^2}\right ) \]