I've been searching all over the internet, but cannot find an answer to the following question:
Given some ODE $y' = f(y, x)$ with initial condition $y(0) = 0$, where $f$ is some real function in two variables, what are the values of the STEP LENGTH, call it $h$, that will ensure stability of the numerical solution obtained by the forward Euler method? The values of $h$ that ensure stability obviously depend on $f$. Is there a simple formula of some sort involving $f$ that allows for (relatively) easy calculation of the values of $h$ for which the numerical solution is stable?
Note: Suppose the solution to the original problem is unique.
The stability region is a circle of radius $1$ around $-1$. This is usually one if the first examples if stability of RK methods is discussed. For practical purposes you should keep $Lh\le 1$ where $L$ is the Lipschitz constant of $f$. This does not guarantee stability, the shape of the circle does not allow such a claim, but should keep the result reasonable for an order 1 method.