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.

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