Creating a Deterministic Push-Down Automaton for the Union of two languages
There is no general algorithm for computing a DPDA $M^*$ as the intersection of two DPDA $M,M'$; that problem is undecidable.  See Undecidable problem intersection of two DCFL languages is DCFL? for a proof.

So, you will have to look for some pattern in this particular problem that makes it easier than the general case.  Hint: what level in the Chomsky hierarchy is $L_1$ at?  what about $L_2$?