13.4.2 problem 4

Internal problem ID [2339]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 1.9. Page 66
Problem number : 4
Date solved : Monday, January 27, 2025 at 05:47:59 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+exp(t*y(t))*(1+t*y(t))+(1+exp(t*y(t))*t^2)*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: 2.786 (sec). Leaf size: 31

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

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