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Separable differential equation

  1. Jun 10, 2004 #1
    I'm doing some revision on differential equations, and am getting stuck on what should be a simple problem.

    The question is:

    Solve: dP/dt = 0.2P(1000 - P)

    Find particular solutions when (i) P(0) = 1000 and (ii) P(0) = 2000

    The answers are supposed to be:

    i) P(t) = 1000
    ii) P(t) = 1000[1 - (1/2)exp(-200t)]^(-1)

    I keep getting the wrong answers. The general solution I came up with (after doing the partial fraction decomposition and integrating) is:

    t + C = 1/200[ ln|P| - ln|1000 - P| ]
    exp(200t + C) = exp(ln|P| - ln|1000 - P|)
    Aexp(200t) = P/(1000 - P) {where A = exp(C)}
    P = 1000Aexp(200t) - (P)Aexp(200t)
    P[ 1 + Aexp(200t) ] = 1000Aexp(200t)

    P = 1000Aexp(200t)/[ 1 + Aexp(200t)]

    But I don't get the right answers when I solve for A with the initial condition.

    I'd appreciate it if anyone can tell me where I'm going wrong here.
  2. jcsd
  3. Jun 10, 2004 #2


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    (Right when you started to solve the diff. eq.)

    ii) Nothing wrong with your answer here, you'll get A=-2 (right?)

    [tex]\frac{1000Ae^{200t}}{1+Ae^{200t}}=\frac{1000}{1+\frac{1}{A}e^{-200t}}, A\neq0[/tex]
  4. Jun 10, 2004 #3
    Aaahh...of course. Thanks, that clears things up a bit.
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