I'm solving a system of stiff ODEs, at first I wanted to implement BDF, but it seem to be a quite complicated method, so I decided to start with Backward Euler method. Basically it says that you can solve an ODE:
$y'=f(t,y)$, with initial guess $y(t_0)=y_0$
Using the following approximation:
$y_{k+1}=y_k+hf(t_{k+1},y_{k+1})$, where $h$ is a step size on parameter $t$
Wikipedia article says that you can solve this equation using Newton-Raphson method, which is basically a following iteration:
$x_{n+1}=x_n-\frac{g(x_n)}{g'(x_n)}$
So, the question is how to correctly mix them together? What initial guess $x_0$ and function $g$ should be?
Also $f$ is quite complex in my case and I'm not sure if it possible to find another derivative of it analytically. I want to write an implementation of it by myself, so predefined Mathematica functions wouldn't work.