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Math Help - Solving system of ODE's

  1. #1
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    Solving system of ODE's

    I have a 2 seperate systems i am trying to solve but don't seem to know any technique that will solve this problem (the other is similar in nature to this one)

    Problem:

    \frac{dC_{0}}{dt} = -\frac{3}{8} C_{0}^3 -\frac{3}{8} C_{1}^2 C_{0}

    \frac{dC_{1}}{dt} = -\frac{3}{8} C_{1}^3 -\frac{3}{8} C_{0}^2 C_{1}

    How would i go about solving this problem??? i have looked everywhere and i can't figure it out!!!!

    Thanks in advance!

    MT
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  2. #2
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    Quote Originally Posted by robotENGR View Post
    I have a 2 seperate systems i am trying to solve but don't seem to know any technique that will solve this problem (the other is similar in nature to this one)

    Problem:

    \frac{dC_{0}}{dt} = -\frac{3}{8} C_{0}^3 -\frac{3}{8} C_{1}^2 C_{0}

    \frac{dC_{1}}{dt} = -\frac{3}{8} C_{1}^3 -\frac{3}{8} C_{0}^2 C_{1}

    How would i go about solving this problem??? i have looked everywhere and i can't figure it out!!!!

    Thanks in advance!

    MT
    Note that each can be written as

     <br />
\frac{d C_0}{dt} = -\frac{3}{8} C_0\left(C_0^2 + C_1^2 \right)<br />
     <br />
\frac{d C_1}{dt} = -\frac{3}{8} C_1\left(C_0^2 + C_1^2 \right)<br />

    so dividing the two gives

     <br />
\frac{d C_0}{C_0} = \frac{d C_1}{C_1}\;\;\; \Rightarrow\;\;\;\ C_1 = k C_0.<br />

    From the first we have

     <br />
\frac{d C_0}{dt} = -\frac{3}{8} C_0\left(C_0^2 + k^2 C_0^2 \right) = -\frac{3}{8} C_0^3\left(1 + k^2 \right)<br />

    which can be solved.
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