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Trouble solving an ordinary differential equation

  1. Jan 12, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the appropriate equation.

    2. Relevant equations

    So there we have the ordrinary differential equation

    [itex]\frac{d M}{dt}=k_1M-k_2(1-M)=A\exp \left (-\frac{E}{T} \right )M-B\exp \left ( -\frac{F}{T} \right )(1-M)[/itex]

    The goal is to solve the differential equation. It turns out the solution should be something like this:

    [itex]M=\frac{k_2}{K_1+k_2}+\frac{k_1}{K_1+k_2}\exp -(k_1+k_2)t[/itex]

    although I think there may be a typo around the last exp (im not sure if t is inside the exponent or not)


    3. The attempt at a solution

    After integrating over t I get

    [itex]M=Mt(k_1+k_2)-k_2 t[/itex]

    But I'm not even sure this is the correct integral of the equation as I don't know how the supposed solution follows from this.
     
  2. jcsd
  3. Jan 12, 2012 #2

    lanedance

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    Homework Helper

    You integration over t is not correct, you need to remember M=M(t) is a function of t.

    To integrate directly the DE must be separable, this one is not but i think it can be made so with a simple substitution
     
  4. Jan 12, 2012 #3

    lanedance

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    An example of a separable DE is as follows
    [tex]
    \frac{dx}{dt} = kx
    [/tex]

    rearranging and integrating gives
    [tex]
    \int\frac{dx}{x} = k\int dt
    [/tex]
    [tex]
    ln(x) = kt+c
    [/tex]
    [tex]
    x = e^{c}e^{kt}
    [/tex]
     
  5. Jan 12, 2012 #4
  6. Jan 12, 2012 #5

    lanedance

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    or write your ODE as
    M' = aM+b

    and make the subsitution
    N = aM+b
     
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