Delay Constrained Shortest Path in $O(D \cdot |E|)$ time
Hint: For $1 \leq d \leq D$, compute inductively the shortest distance between $s$ and any other node at delay at most $d$. Each such iteration runs in $O(|E|)$, so the total running time is $O(D|E|)$. (This assumes that the graph is connected. Otherwise you have to replace $|E|$ with $|E|+|V|$.)