Determine the nature and stability of the critical point (0,0) for the following system: d x d t = x + 2 y + 2 sin ⁡ y d y d t = − 3 y − x e x

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Determine the nature and stability of the critical point (0,0) for the following system:            d      x              d      t        =  x  +  2  y  +  2  sin  ⁡  y            d      y              d      t        =  −  3  y  −  x      e          x

Jeroronryca · Accepted Answer

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) = 0g(x,y) = 0Now  = 0  ⇒  x  +  2  y  +  2  sin  ⁡  y  =  0and =0Therefore   −  3  y  −  x      e          x        =  0From these two we observe that the critical point is (0,0)

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

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