\section{(The Reality of Absolute Stability)}
Here we apply Euler's Method to the IVP
\[ \begin{array}{lll}
	y' = -100(y - \sin{x}) + \cos{x}, & y(0) = 0, & x \in [0,20]
\end{array}\]
The exact solution of this problem is $y(x) = \sin{x}$.  To find the smallest value of $h$ that will make this method unstable, we look at the stability region of Euler's method.  We know that the stability region of Euler's method is defined as $e^{i\theta} - 1$, which is the unit circle shifted to the left.  And we know, that for this problem, $\lambda = -100$.  Since $\lambda$ is real, we are only considering the real line on the interval $(-2,0)$.  Then if we solve for the left end point, we see that $h = 1/50$.

See attached image for a plot of the error $e = y(20) - y_{n}$ against $h$.  Notice that for values of $h$ smaller than specified by this method, we get very unstable results.
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