Question

Determine the steady state of the following ODE system, where a and b are assumed constants allowed in the solution:

$dx/dt=0 = -x +a*y+x^2*y$

$dy/dt = 0 = b-a*y-x^2*y$

Please show all steps clearly.

Answer 1

Given equations are dx/dt = -x + ay + x^2 y dy/dt = b - ay - x^2 y for steady state both equation implies 0 so, -x + ay + x^2 y = 0 implies (x^2 + a) y = x implies y = x/x^2 + a putting this volume of y in 11 b - a x/x^2 + a - x^2 x/x^2 + a = 0 implies b(x^2 + a) - ax - x^3 = 0 implies x^3 + ax = bx^2 + ab implies x^3 - bx^2 + ax - ab = 0 implies x^2 (x - b) + a (x - b) = 0 implies (x - b) (x^2 + a) = 0 implies x = b, x = squareroot - a \r\n now roots of x will always be real for steady state, so x^2 + a = 0 is not a seaside's solution for this equation Hence (x - b) = 0 is the only steady p for this system y = x/x^2 + a = b/b^2 + a