CMOS inverter : Calculation of Vd
This is a homework problem, right?

First of all check whether transistor conduct. 


PMOS conducts when \$V_{SG}&gt;|V_T|\$.
NMOS conducts when \$V_{GS}&gt;V_T\$.


If any of them is in cut-off, then the solution is very simple, right? If both are not in cut-off then we proceed.

You got two separate transistors. This means that you can write two separate current equations. This complicates by the fact that MOSFETs have different equations for "linear" and "saturation" modes of operation. There are four possible combinations for modes of PMOS-NMOS: linear-linear, linear-saturation, saturation-linear, saturation-saturation. You'll need to guess here and check if your guess works - if it doesn't, then you'll get unsolvable equations. In this case you simply check the other combinations. However, just by inspection of the circuit you may be able to eliminate some of the possibilities.

What the unknowns are in these equations? Hint: there are up to four of them (currents, voltages).

You can't solve for four unknowns having just two equations. Too bad. But if you can relate some of the unknowns to each other, you may be able to reduce their number. Try to see what constraints are imposed by the schematic (current relations, voltage relations, ...).

The combination of modes which provides you with a solution for \$V_d\$ is the answer (together with the solution itself).