1.52 problem 52

Internal problem ID [7744]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 52
Date solved : Tuesday, October 22, 2024 at 02:29:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

\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*}

1.52.1 Maple step by step solution

1.52.2 Maple trace

Methods for second order ODEs:
 

1.52.3 Maple dsolve solution

Solving time : 0.015 (sec)
Leaf size : 80

dsolve(diff(diff(y(t),t),t)+(t^2-1)/t*diff(y(t),t)+t^2/(1+exp(1/2*t^2))^2*y(t) = 0, 
       y(t),singsol=all)
 

\[ y = \frac {\left (c_1 \left ({\mathrm e}^{\frac {t^{2}}{2}}\right )^{\frac {i \sqrt {3}}{2}} \left (1+{\mathrm e}^{\frac {t^{2}}{2}}\right )^{-\frac {i \sqrt {3}}{2}}+c_2 \left ({\mathrm e}^{\frac {t^{2}}{2}}\right )^{-\frac {i \sqrt {3}}{2}} \left (1+{\mathrm e}^{\frac {t^{2}}{2}}\right )^{\frac {i \sqrt {3}}{2}}\right ) \sqrt {1+{\mathrm e}^{\frac {t^{2}}{2}}}}{\sqrt {{\mathrm e}^{\frac {t^{2}}{2}}}} \]

1.52.4 Mathematica DSolve solution

Solving time : 0.141 (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]
 

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