Exercise 3

A system is autonomous if $dx/dt$ and $dy/dt$ do not depend on $t$; that is, $$\begin{aligned} \frac{dx}{dt} & = f(x,y) \\ \frac{dy}{dt} & = g(x,y). \end{aligned}$$ Which of the following systems are autonomous?

1. $\dfrac{dx}{dt} = 4x -3y; \dfrac{dy}{dt} = 3x + 4y$

2. $\dfrac{dx}{dt} = 4x -3y + \cos x; \dfrac{dy}{dt} = 3x + 4y - \sin x$

3. $\dfrac{dx}{dt} = 4x -3y +3tx; \dfrac{dy}{dt} = 3x + 4y - e^x$

4. $\dfrac{dx}{dt} = 4x; \dfrac{dy}{dt} = 4y$