# Determine the nature and stability of the critical point (0,0) for the following system: dx/dt =x+2y+2 sin y dy/dt =-3y-xe^x

Determine the nature and stability of the critical point (0,0) for the following system:
$\frac{dx}{dt}=x+2y+2\mathrm{sin}y$
$\frac{dy}{dt}=-3y-x{e}^{x}$
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Jeroronryca
The equilibrium solutions (or points) to a system of first order differential equations are the points at which the first derivatives are equal to zero.
That is, for the system:
dx/dt = f(x,y)
dy/dt = g(x,y),
the equilibrium points are the solutions to the algebraic equations:
f(x,y) = 0
g(x,y) = 0
Now = 0
$⇒x+2y+2\mathrm{sin}y=0$
and =0
Therefore $-3y-x{e}^{x}=0$
From these two we observe that the critical point is (0,0)