I have the following system of differential equations:
$dy_i / dx = c_i(y_{i-1}^4 - y_{i}^4) +f_i$ (for i=1,...,n)
that I am desperately trying to express as follows (using Euler's method):
$\textbf{y}(t_{k+1})=\boldsymbol{C_1}\boldsymbol{y}(t_{k})+\boldsymbol{C_2}\boldsymbol{y^4}(t_{k}) + \boldsymbol{f}$
Where $C_1$ and $C_2$ are transition matrices (i.e. stochastic matrices) in the finite difference equations. I have tried expressing as (where y = (y1, y2)'):
$\frac{d\boldsymbol{y}}{dt} = \boldsymbol{F}(t,\boldsymbol{y})=\begin{pmatrix} -c_1y_1^4\\ c_2(x^4-y^4)\end{pmatrix}$
However this does not produce the equation in the desired form (which is necessary for a simulation technique I will subsequently apply). Note $f_i = 0$ for all i except $i=1$, where it's equal to 1. We may assume $y_1(0)=1$ and $y_2(0)=0$. The idea is taken from: https://www.witpress.com/Secure/elibrary/papers/CE99/CE99014FU.pdf (see the two equations at the bottom of page 6).
I am really stuck with this one, so any help would be highly appreciated.