Treating the derivative as a linear operator for a system modeled by a differential equation?

Question

I have a system with input $x(t)$ and output $y(t)$.

It is modeled by this differential equation: $\dfrac{d^2y(t)}{dt^2} + y(t) = \dfrac{dx(t)}{dt} + x^2(t)$

I need to see if this system is linear and time invariant. I treat the derivative as a linear operator and: $y(t) = \dfrac{D\times x(t)+x^2(t)}{D^2 + 1}$.

Then, I can test for time invariance and linearity by doing the usual method. Can I do this? Why or why not? It happens to work in this case, but I don't know for all cases.