74.6.35 problem 36

Internal problem ID [16058]
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 : Tuesday, January 28, 2025 at 08:32:08 AM
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*}

✓ Solution by Maple

Time used: 0.335 (sec). Leaf size: 18

dsolve([(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,y(0) = 0],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 60.429 (sec). Leaf size: 20

DSolve[{(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,{y[0]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to W\left (-\frac {t^2}{e^{t^2}+1}\right ) \]