# Thread: Solving system of ODE's

1. ## 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:

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

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

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

MT

2. Originally Posted by robotENGR
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:

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

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

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

MT
Note that each can be written as

$\displaystyle \frac{d C_0}{dt} = -\frac{3}{8} C_0\left(C_0^2 + C_1^2 \right)$
$\displaystyle \frac{d C_1}{dt} = -\frac{3}{8} C_1\left(C_0^2 + C_1^2 \right)$

so dividing the two gives

$\displaystyle \frac{d C_0}{C_0} = \frac{d C_1}{C_1}\;\;\; \Rightarrow\;\;\;\ C_1 = k C_0.$

From the first we have

$\displaystyle \frac{d C_0}{dt} = -\frac{3}{8} C_0\left(C_0^2 + k^2 C_0^2 \right) = -\frac{3}{8} C_0^3\left(1 + k^2 \right)$

which can be solved.