14.3.2 problem 4

Internal problem ID [2511]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 1. First order differential equations. Section 1.9. Exact equations. Excercises page 66
Problem number : 4
Date solved : Monday, January 27, 2025 at 05:56:37 AM
CAS classification : [_exact]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 32

dsolve((1+(1+t*y(t))*exp(t*y(t)))+(1+t^2*exp(t*y(t)))*diff(y(t),t)=0,y(t), singsol=all)
 

\[ y = \frac {-c_1 t -t^{2}-\operatorname {LambertW}\left (t^{2} {\mathrm e}^{-t \left (t +c_1 \right )}\right )}{t} \]

✓ Solution by Mathematica

Time used: 3.173 (sec). Leaf size: 31

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

\[ y(t)\to -\frac {W\left (t^2 e^{t (-t+c_1)}\right )}{t}-t+c_1 \]