D to JK and JK to D conversion

D flip-flop into JK flip-flop

Find D (the input of the available flip-flop) for each J, K, $Q_n$. D excitation: $D = Q_{n+1}$.

        KQn=00 01 11 10
J=0        0   1  0  0
J=1        1   1  0  1

$$D = J\,Q_n' + K'\,Q_n$$

J --[AND]--+
Q'--[   ]  +--[OR]-- D +--------+--- Q
K'--[AND]--+           | D FF   |
Q --[   ]              |  CLK   |--- Q'
                       +--------+

JK flip-flop into D flip-flop

Find J, K for each D, $Q_n$, using the JK excitation table.

From the table (using don't cares): $J = D$, $K = D'$.

D --+------------ J +--------+--- Q
    |               | JK FF  |
    +--[NOT]----- K |  CLK   |--- Q'
                    +--------+

So D goes to J directly and through an inverter to K. Then $D = 1$ sets and $D = 0$ resets the flip-flop at each clock, which is D flip-flop behaviour.