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, March 30, 2025 at 01:30:43 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

✓ 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);
 

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

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 coef\ 
ficients was determined
 

✓ 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]
 

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

✓ 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)
 

$$y\left(t\right)=C_2{t}^2(1-\frac{t^2}{4})+C_1+O\left(t^6\right)$$