68.6.35 problem 36

Internal problem ID [17345]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.4, page 57
Problem number : 36
Date solved : Thursday, October 02, 2025 at 02:06:13 PM
CAS classification : [_exact]

\begin{align*} 2 t y \,{\mathrm e}^{t^{2}}+2 t \,{\mathrm e}^{-y}+\left ({\mathrm e}^{t^{2}}-t^{2} {\mathrm e}^{-y}+1\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.211 (sec). Leaf size: 18

ode:=2*t*y(t)*exp(t^2)+2*t*exp(-y(t))+(exp(t^2)-t^2*exp(-y(t))+1)*diff(y(t),t) = 0; 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \operatorname {LambertW}\left (-\frac {t^{2}}{{\mathrm e}^{t^{2}}+1}\right ) \]

✓ Mathematica. Time used: 60.288 (sec). Leaf size: 20

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

\begin{align*} y(t)&\to W\left (-\frac {t^2}{e^{t^2}+1}\right ) \end{align*}

✗ Sympy

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

Timed Out