TITLE: How do I get the $f(t_{n+1}, y_{n+1})$ needed to use the implicit Euler method?
QUESTION [6 upvotes]: I have to solve a system of 1st order ODEs which are implicit.
I know the formula for Explicit or forward Euler method is:
$$y_{n+1}= y_n + hf(t_n, y_n),$$
whereas the formula for implicit or backward Euler method is
$$y_{n+1}= y_n + hf(t_{n+1}, y_{n+1}).$$
In order to use the implicit Euler method, how can i get the value of $f(t_{n+1}, y_{n+1})$ ?
Can I use the forward Euler method to get the value $y_{n+1}$ then substitute in backward Euler formula?

REPLY [2 votes]: You can use fixed point iteration or Newton iteration
Fixed point iteration is
$$y_{n+1}^{(s+1)}= y_n + hf(t_{n+1}, y_{n+1}^{(s)})$$
with iteration starting with explicit Euler: $y_{n+1}^{(0)}= y_n + hf(t_{n+1}, y_n)$