Determine the nature and stability of thecritical point (0,0) for the following system: d x d t = − sin ⁡ ( x − y ) d y d t = 1 − 5 y − e x

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Determine the nature and stability of thecritical point (0,0) for the following system:            d      x              d      t        =  −  sin  ⁡  (  x  −  y  )            d      y              d      t        =  1  −  5  y  −      e          x

dominicsheq8 · Accepted Answer

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) = 0g(x,y) = 0 = 0This implies that   x  −  y  =  0  ⇒  x  =  y=0Then the above Equation becomes  1  −  5  x  −      e          x        =  0  ⇒      e          x        =  1  −  5  xwhen x = 0 both L.H.S and R.H.S are EqualTherefore x = 0 and y = 0Therefore the critical point is (0,0)

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

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