Upper bound for affine differential equation

Question

Let $\frac{dx}{dt} = a x + b$ be a stable affine differential equation where $a \in \mathbb{R}^-,b \in \mathbb{R}$ and let $c \in \mathbb{R}^-, d \in \mathbb{R}^+$. How can we determine a maximum upper bound $u(x)$ such that $x$ remains inside $[c,d]$ if initially $x_0 \in [c,d]$ and $b \leq u(x_0)$?