56.1.52 problem 52

Internal problem ID [8764]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 52
Date solved : Tuesday, January 28, 2025 at 03:16:45 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 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 \end{align*}

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 80

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)
 

\[ y = \frac {\left (c_{1} \left (1+{\mathrm e}^{\frac {t^{2}}{2}}\right )^{-\frac {i \sqrt {3}}{2}} \left ({\mathrm e}^{\frac {t^{2}}{2}}\right )^{\frac {i \sqrt {3}}{2}}+c_{2} \left (1+{\mathrm e}^{\frac {t^{2}}{2}}\right )^{\frac {i \sqrt {3}}{2}} \left ({\mathrm e}^{\frac {t^{2}}{2}}\right )^{-\frac {i \sqrt {3}}{2}}\right ) \sqrt {1+{\mathrm e}^{\frac {t^{2}}{2}}}}{\sqrt {{\mathrm e}^{\frac {t^{2}}{2}}}} \]

✓ Solution by Mathematica

Time used: 0.148 (sec). Leaf size: 72

DSolve[D[y[t],{t,2}]+(t^2-1)/t*D[y[t],t]+t^2/(1 + Exp[t^2/2])^2*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^{\text {arctanh}\left (2 e^{\frac {t^2}{2}}+1\right )} \left (c_2 \cos \left (\sqrt {3} \text {arctanh}\left (2 e^{\frac {t^2}{2}}+1\right )\right )-c_1 \sin \left (\sqrt {3} \text {arctanh}\left (2 e^{\frac {t^2}{2}}+1\right )\right )\right ) \]