Boolean expressions from Bubble Diagram for D-flip flop entries
Consider D0 and D1 as two separate signals, each of which is determined by 3 inputs (D0, D1, and x). Create two separate Karnaugh maps for the two signals, and determine what logic is necessary to create the next value of D0 and D1 given the current state of D0, D1, and x.

As you have observed, it's possible that you end up with trivial logic for some of the bits.