Internal problem ID [7096]
Book: Own collection of miscellaneous problems
Section: section 1.0
Problem number: 52.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\frac {\left (t^{2}-1\right ) y^{\prime }}{t}+\frac {t^{2} y}{\left (1+{\mathrm e}^{\frac {t^{2}}{2}}\right )^{2}}=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 84
dsolve(diff(y(t),t$2)+(t^2-1)/t*diff(y(t),t)+t^2/(1 + exp(t^2/2))^2*y(t)=0,y(t), singsol=all)
\[ y \left (t \right ) = \frac {\left (c_{1} \left (1+{\mathrm e}^{\frac {t^{2}}{2}}\right )^{-\frac {i \sqrt {3}}{2}} \left ({\mathrm e}^{\frac {t^{2}}{2}}\right )^{\frac {i \sqrt {3}}{2}}+c_{2} \left (1+{\mathrm e}^{\frac {t^{2}}{2}}\right )^{\frac {i \sqrt {3}}{2}} \left ({\mathrm e}^{\frac {t^{2}}{2}}\right )^{-\frac {i \sqrt {3}}{2}}\right ) \sqrt {1+{\mathrm e}^{\frac {t^{2}}{2}}}}{\sqrt {{\mathrm e}^{\frac {t^{2}}{2}}}} \]
✓ Solution by Mathematica
Time used: 0.116 (sec). Leaf size: 72
DSolve[y''[t]+(t^2-1)/t*y'[t]+t^2/(1 + Exp[t^2/2])^2*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to e^{\text {arctanh}\left (2 e^{\frac {t^2}{2}}+1\right )} \left (c_2 \cos \left (\sqrt {3} \text {arctanh}\left (2 e^{\frac {t^2}{2}}+1\right )\right )-c_1 \sin \left (\sqrt {3} \text {arctanh}\left (2 e^{\frac {t^2}{2}}+1\right )\right )\right ) \]