I have a set of coupled ODEs:
\begin{align} \dot{x} &= -A x^4 + B_1 y^4, \\ \dot{y} &= A x^4 - (B_1 + B_2) y^4 + C z^4, \\ \dot{z} &= B_2 y^4 - C z^4 \end{align}
Without having to use the first one, I've managed to decouple them to get an expression involving only z and x: I rearranged the third equation to get another expression for $\dot{y}$. Subbing this into the middle equation gives
$$ \dot{y} = \frac{1}{4} \left[ \frac{\dot{z} + Cz^4}{B_2} \right]^{-\frac{3}{4}} \left[ \frac{\ddot{z} + 4Cz^3\dot{z}}{B_2} \right] = Ax^4 - \frac{B_1 + B_2}{B_2} ( \dot{z} + Cz^4 ) + Cz^4 $$
However this is a horrible looking nonlinear ODE that I have no idea how to tackle.
Is there a better approach that I'm missing, or will I have to resort to numerical approaches?
Tagged physics since this is a model of heat transfer by radiation between three objects (x,y,z are temperatures).
Answering my own question, although I'd love to be proved wrong!
I've ended up just going for a numerical approach with this one. In case anyone is interested, this was a model for radiative heat transfer betwen three isolated objects with temperatures x, y and z. x and z only couple to y, whereas y couples to both the other.
As for a solution, I ended up adding a fourth layer which I held constant as a perfect heat-sink. Here's what happens if you step the perfect heat-sink's temperature. In the following plot x,y,z have become "chamber," "shield" and "cavity" respectivly.
And in the frequency domain:
So there you go. I'd still love to see an analytic solution if you fancy having a try!