Edit distance where insert costs 1, delete costs 2 and replace costs 3. Define the DP state and the recurrence, and tell me what changes from the classic version.
The state is unchanged - the minimum cost to transform the first i characters of one string into the first j of the other - because the subproblem structure does not depend on what operations cost. Only the recurrence changes, adding a per-operation weight instead of a uniform 1, and the base cases become multiples of the insert and delete costs rather than i and j.