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Initial value problem

  1. Jul 18, 2007 #1
    1. The problem statement, all variables and given/known data
    Hello, I am wondering if someone can look at the following:

    suppose i'm given an initial value problem: dy/dt = ty(4-y)/3 with y(0) = a.

    So, if I separable the variables, i would get:

    dy/4(4-y) = t/3dt
    and I would get by partial fractions
    (1/4)*lny-(1/4)ln(4-y) = t^2/6 + C
    (1/4)*ln(y/4-y) = t^2/6+C
    or (y/4-y) = e^((2/3)t^2 +4C)

    so can anyone tell me how I would determine how the value of the solution would vary depending on the initial value a? in other words, how could i get this into a nice form y = ....

    Thanks very much.
  2. jcsd
  3. Jul 18, 2007 #2


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    I presume that is a typo: dy/(y(4-y))= (t/3)dt

    Last edited: Jul 19, 2007
  4. Jul 18, 2007 #3


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    You can write the solution a little more nicely as y/(4-y)=D*exp((2/3)t^2) (where D=exp(4C) - but it's still just a constant). So your initial value problem is a/(4-a)=D. Solve for a. Clear the fractions, move all the a's to one side etc. You can solve for y(t) in the same way.
  5. Jul 18, 2007 #4
    Got it. Many thanks!
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