solving differential equation help

$\displaystyle \frac{dp}{dt} = bp(p-a)+c-fp^{3}, p(0) = p_{0} $

use the parameter values a=-3/4, b = 42, c =-81, f=10

i) Find the steady states of the differential equation and determine wheather each one is linearly stable or unstable.

ii) Use the modified parameter vales a=23/28, b=42, c=8, f=10. Determine $\displaystyle \lim_{t\to\infty}p(t) $ for all initial values $\displaystyle p_{0} \geq 0 $

I have worked out the steady state solution to the differential equation, by setting it equal to zero

$\displaystyle -40p^{3} + 168p^{2} + 126p-324 = 0 $

on matlab I get

$\displaystyle p = \frac{6}{5} , \frac{-3}{2} , \frac{9}{2} $

I am however stuck on the second part of this question, dont know how to go about it.?

Any help appreciated.

Thank you.

Re: solving differential equation help

Hey Tweety.

Hint: For what values of p0 does the function converge to the steady state? (You have three steady state solutions and the derivative does not depend on time, so look at when p is to the left and to the right of steady state solutions and if/when they approach them).

Re: solving differential equation help

Quote:

Originally Posted by

**chiro** Hey Tweety.

Hint: For what values of p0 does the function converge to the steady state? (You have three steady state solutions and the derivative does not depend on time, so look at when p is to the left and to the right of steady state solutions and if/when they approach them).

Thank you, however I am not sure how to work out for what values of p0 does the function converge to the steady states?

Do I just choose random number for p0? and input into the equation?

Re: solving differential equation help

The key thing is where the derivative is positive and negative.

For example if the derivative was positive after the last steady state, then it would shoot off to infinity with an increase in time. If however it was negative, it would converge to the steady state solution.

This is the kind of thing you have to look at.

Re: solving differential equation help

so should I draw a graph of the equation ?