Given the logistic equation dy/dt=2y(1-y/5)

Question

a.) Determine the general solution of this equation.

b.) FInd the particular solution satisfying y(0)=3

c.) Evaluate lim y as t->infinity and explain its meaning.

Answer 1

a) Rewriting

5y' =2y(5-y)

We can use separation of variables

5y'/(y(y-5)) = 2

Integrating, the right side becomes 2t + C

Looking at the left side using partial fractions

∫5/(y(5-y))dy = ∫1/y + 1/(5-y) dy = ln(y)-ln(5-y) = ln(y/(5-y))

Putting both sides together and taking to the power of e

y/(5-y) = e^2t+C

Separating out y

y = 5e^2t+C/(1+e^2t+C)

b) y(0) = 3 = 5e^C/(1+e^C) --> C = ln(3/2)

y = 5(3/2)e^2t/(1+(3/2)e^2t) = 15e^2t/(2+3e^2t)

c) lim t→∞ y = 15/3 = 5

The most y can be is 5