Trying to find steady state
Umm, this question can be generalized. Here Vin is 0V, but if Vin is other values, you can think as below.

No matter which region the transistor working, the charging current to the capacitor, that is \$i_{D}\$, should tend to 0. Then we can do some analysis on the \$i_{D}\$'s equation on these two regions.

At nonsaturation region

$$
v_{DS} &lt; v_{GS}-v_{TN}\\
i_{D}=K_{n}[2(v_{GS}-V_{TN})v_{DS}-v_{DS}^2]
$$ 

When \$i_{D} = 0\$, this equation has only one solution, that is \$v_{DS} = 0\$.

At saturation region

$$
v_{DS} &gt; v_{GS} - v_{TN}\\
i_{D}=K_{n}(v_{GS}-V_{TN})^2
$$

When \$i_{D}=0\$, there is \$v_{GS}=V_{TN}\$.

That is, in ideal case (omit the leakage), the capacitor is either charged to \$v_{D}\$ or charged to \$v_{G}-V_{TN}\$.