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Homework Help: Limits of Differential Equations

  1. Jan 28, 2015 #1
    1. The problem statement, all variables and given/known data
    I need help finding the limit of the differential equation.
    (dx/dt) = k(a-x)(b-x) that satisfies x(0)=0
    a) 0<a<b and find the limit as t->infinity of X(t)
    b) 0<a=b and find the limit as t->infinity of X(t)

    2. Relevant equations

    3. The attempt at a solution

    I separated the equation in part a and attempted to solve for x and got a nasty equation
    http://www4b.wolframalpha.com/Calculate/MSP/MSP115222ac5cd5fd1ghhf0000016hfg120ch979ad9?MSPStoreType=image/gif&s=20&w=156.&h=41. [Broken] then I solved for c and found it to be c=-(a/b). I plugged that in for c and got:
    http://www4f.wolframalpha.com/Calculate/MSP/MSP49220eh2769a9a2d53700001g9fiib9hd1eh2c3?MSPStoreType=image/gif&s=49&w=159.&h=50. [Broken] I don't know how to take it further.
    I believe that the answer to part a is a based of a graph, but I am unable to prove it.
    Thanks in advance.
    Last edited by a moderator: May 7, 2017
  2. jcsd
  3. Jan 29, 2015 #2


    Staff: Mentor

    Did you check the solution you got for x in your first equation above?
    Last edited by a moderator: May 7, 2017
  4. Jan 29, 2015 #3


    User Avatar
    Homework Helper

    The last equation is wrong. Why did you change the second exponent?

    You can replace ## e^{akt} e^{-bkt } = e^{(a-b)kt } ## in the first equation. The value c=-a/b is right. Just plug in for c.
    Last edited by a moderator: May 7, 2017
  5. Jan 29, 2015 #4
    Never mind I had it right early. Thanks anyway everyone!
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