7) Consider the DE $x' = -rx$ with $r > 0$, which admits the general solution $x(t) = ce^{-rt}$ for $c \in \mathbb{R}$.
a) Write down an expression for $x_{n+1}$ for this DE using Euler's method with step size $h > 0$.
b) The analytical solution satisfies $x(t) \to 0$ as $t \to +\infty$, for any $c \in \mathbb{R}$, and the larger the value of $r$, the faster the decay toward zero. What condition must $h$ satisfy in order to ensure that $x_n$ approaches zero as $n$ becomes large? (Hint: Simplify your answer from part (a) by writing $x_{n+1} = (\text{expression involving } r, h)x_n$ and use this to determine what $r$ and $h$ must satisfy so that $|x_{n+1}| < |x_n|$, guaranteeing that $x_n \to 0$.)
