= Solution
The <amplification factor> is $R=1-\lambda/N$. To reproduce decay, the <forward Euler method> requires $|R|<1$, equivalently
$$
-1<1-\frac\lambda N<1
\quad\Longleftrightarrow\quad
\boxed{N>\lambda/2.}
$$
At $N=\lambda/2$, $R=-1$ and the approximations alternate with constant magnitude $2$; for $N<\lambda/2$, $|R|>1$ and the magnitude grows. This is the condition of <absolute stability> for this decaying <differential equation>.
Stability is only a minimum requirement. Values just inside the bound can still give a poor endpoint approximation; resolving the decay accurately generally requires a smaller <step size>. If positivity at every step is desired, the stronger condition $N\ge\lambda$ ensures $R\ge0$. The figure shows the difference between decay of the exact solution and numerical oscillation.
\Image[/past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/discrete-main-2015-euler-stability.png]
{title=Forward Euler approximation to exponential decay: instability, neutral oscillation, and stable decay}
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