56.1.52 problem 52

Internal problem ID [8764]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 52
Date solved : Sunday, March 30, 2025 at 01:30:43 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*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 80

ode:=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; 
dsolve(ode,y(t), singsol=all);
 

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

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

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

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

✓ Sympy. Time used: 1.835 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**2*y(t)/(exp(t**2/2) + 1)**2 + Derivative(y(t), (t, 2)) + (t**2 - 1)*Derivative(y(t), t)/t,0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{2} t^{2} \left (1 - \frac {t^{2}}{4}\right ) + C_{1} + O\left (t^{6}\right ) \]