Question

Find an autonomous differential equation with all of the following properties:

equilibrium solutions at y=0 and y=3,
y' > 0 for 0 y' < 0 for -inf < y < 0 and 3 < y < inf
dy/dx =

Answer 1

dy/dt = y ( 3 - y )

Explanation:

Given data:

Determine an autonomous differential equation with the following properties

y = 0 and Y = 3

y' > 0 for 0 < y < 3

y' < 0 for -∞ < y < 0 and 3 < y < ∞

considering an autonomous differential equation

dy/dt = y ( 3 - y )

y = 0 and 3 represents equilibrium solutions

if 0 < y < 3 then y ( 3 - y ) > 0 for 0 < y < 3

hence : dy / dt = y' > 0 for 0 < y < 3

y ( 3 - y ) < 0 for -∞ < y < 0 and 3 < y < ∞

hence : dy / dt = y' < 0 for -∞ < y < 0 and 3 < y < ∞

this shows that the autonomous differential equation satisfies every condition hence the autonomous differential equation is :

dy/dt = y ( 3 - y )