2.1.52 Problem 52

Internal problem ID [8764]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 52
Date solved : Sunday, February 23, 2025 at 04:37:27 AM
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*}

Maple step by step solution

Maple trace

`Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius 
-> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
-> Trying changes of variables to rationalize or make the ODE simpler 
<- unable to find a useful change of variables 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   trying 2nd order exact linear 
   trying symmetries linear in x and y(x) 
   trying to convert to a linear ODE with constant coefficients 
   <- to_const_coeffs successful: conversion to a linear ODE with constant coefficients was determined`
 

Maple dsolve solution

Solving time : 0.037 (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)
 

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

✓Mathematica DSolve solution

Solving time : 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]
 

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

✓Sympy solution

Solving time : 1.835 (sec)
Leaf size : 19

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

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)
 

Eq(y(t), C2*t**2*(1 - t**2/4) + C1 + O(t**6))
 

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