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Math Help - solving differential equation help

  1. #1
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    solving differential equation help

     \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  \lim_{t\to\infty}p(t) for all initial values  p_{0} \geq 0

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

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

    on matlab I get

     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.
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  2. #2
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    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).
    Thanks from Tweety
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  3. #3
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    Re: solving differential equation help

    Quote Originally Posted by chiro View Post
    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?
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  4. #4
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    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.
    Thanks from Tweety
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  5. #5
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    Re: solving differential equation help

    so should I draw a graph of the equation ?
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