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 : Sunday, March 30, 2025 at 12:07:09 AM
CAS classification : [_exact]

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

With initial conditions

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

✓ Maple. Time used: 0.556 (sec). Leaf size: 36

ode:=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; 
ic:=y(0) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

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

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

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

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

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(2*t + (t*exp(t*y(t))*cos(2*t) - 3)*Derivative(y(t), t) + y(t)*exp(t*y(t))*cos(2*t) - 2*exp(t*y(t))*sin(2*t),0) 
ics = {y(0): 0} 
dsolve(ode,func=y(t),ics=ics)
 

Timed Out