# Numerical Stability of Euler's Method

Calculus Level 5

Suppose we want to use Euler's Method to numerically (discretely) compute the time-equation below through a sequence of iterative steps:

$\large{y = e^{-A t} \\ y_k = y_{k-1} + y'_{k-1} \Delta t}$

In the above equation, $$\Delta t = \frac{1}{1000}$$. $$y_k$$ denotes the present value of the function, $$y_{k-1}$$ denotes the previous value of the function, and $$y'_{k-1}$$ denotes the previous value of the function's time-derivative. The parameter $$A$$ is an arbitrary positive constant.

Suppose $$y_0 = 1$$ and $$y'_0 = -A$$.

For $$0 < A < A_1$$, the system undergoes monotonic decay toward zero.
For $$A_1 < A < A_2$$, the system undergoes damped oscillation.
For $$A > A_2$$, the system undergoes divergent oscillation.

Determine $$A_1 + A_2$$. Assume no rounding or truncation errors (we have a perfect computer).

Note: This version of Euler's Method is more specifically known as the "forward" or "explicit" version.

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