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Find an autonomous differential equation with all of the following properties:Equilibrium solutions at y=0 and y=3, y' > 0 for 0 < y < 3; and y' < 0 for -inf < y < 0 and 3 < y < inf.

Respuesta :

Answer:

dy = 2(c-3y)dt

Step-by-step explanation:

Given the Equilibrium solutions at y=0 and y=3, y' > 0 for 0 < y < 3; and y' < 0 for -inf < y < 0 and 3 < y < inf.

from the boundary conditions given,

  • y' > 0 for 0 < y < 3;
  • y' < 0 for -inf < y < 0
  • and 3 < y < inf

  • since our task is to find the differential of y wrt t i.e dy/dt,
  • from the first condition, it implies that if we are to assume from the range of values of y = {0,1,2}, assume when y =0, t = 0

from y = 3, y-3 = 0

integrate wrt dy i.e Integral (y-3) dy = 0

y2/2 - 3y + c = 0, where c is the constant of integration

hence, y2-6y+2c = 0

from the equation, above for the differential of y (dy/dx) to be greater than zero,  for the boundary conditions, 0 < y < 3, the constant of integrative must be negative or zero.

hence the equation y2-6y+2c = 0 can be written as

dy/dx =2c-6y, dy = 2(c-3y)dt

in terms of t

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