14.3.8 problem 10

Internal problem ID [2517]
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 : 10
Date solved : Monday, January 27, 2025 at 05:57:39 AM
CAS classification : [_exact]

\begin{align*} 2 t -2 \,{\mathrm e}^{y t} \sin \left (2 t \right )+{\mathrm e}^{y t} \cos \left (2 t \right ) y+\left (-3+{\mathrm e}^{y t} t \cos \left (2 t \right )\right ) y^{\prime }&=0 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.376 (sec). Leaf size: 36

dsolve([(y(t)*cos(2*t)*exp(t*y(t))-2*sin(2*t)*exp(t*y(t))+2*t)+(t*cos(2*t)*exp(t*y(t))-3)*diff(y(t),t)=0,y(0) = 0],y(t), singsol=all)
 

\[ y = \frac {t^{3}-3 \operatorname {LambertW}\left (-\frac {t \cos \left (2 t \right ) {\mathrm e}^{\frac {t \left (-1+t \right ) \left (t +1\right )}{3}}}{3}\right )-t}{3 t} \]

✓ Solution by Mathematica

Time used: 4.616 (sec). Leaf size: 43

DSolve[{(y[t]*Cos[2*t]*Exp[t*y[t]]-2*Sin[2*t]*Exp[t*y[t]]+2*t)+(t*Cos[2*t]*Exp[t*y[t]]-3)*D[y[t],t]==0,{y[0]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {t^3-3 W\left (-\frac {1}{3} e^{\frac {1}{3} t \left (t^2-1\right )} t \cos (2 t)\right )-t}{3 t} \]