Using the linear operator to check for time invariance of a differential equation?
Let $y(t)=\mathcal{H}(x(t))$ describe the system by relating its input $x(t)$ to the output $y(t)$. Then, to check linearity simply check with $x(t)=1$ (i.e. a constant). You will see, that 

$$\mathcal{H}(x(t)=1)+\mathcal{H}(x(t)=1)\neq \mathcal{H}(x(t)=1+1)$$

which is one requirement for the linearity. So, the system is non-linear. Actually, you can immediately see this from the fact that $x(t)$ counts quadratically, which is a non-linear relation.

Regarding time-invariance, you see that the time-argument is the same for all functions and the time-value is not used in the equation itself (such as $t\cdot x(t)$ would ne not time-invariant). So, the system is time-invariant.