# Determine the nature and stability of thecritical point (0,0) for the following system: dx/dt =-sin (x-y) dy/dt =1-5y -e^x

Determine the nature and stability of thecritical point (0,0) for the following system:
$\frac{dx}{dt}=-\mathrm{sin}\left(x-y\right)$
$\frac{dy}{dt}=1-5y-{e}^{x}$
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dominicsheq8
The equilibrium solutions (or points) to a system of first order differential equations are the points at which the first derivatives are equalto 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
= 0
This implies that $x-y=0⇒x=y$
=0
Then the above Equation becomes
$1-5x-{e}^{x}=0$
$⇒{e}^{x}=1-5x$
when x = 0 both L.H.S and R.H.S are Equal
Therefore x = 0 and y = 0
Therefore the critical point is (0,0)