Using the linear operator to check for time invariance of a differential equation?

Question

I have a differential equation $$\frac{d^2y(t)}{dt^2}+y(t) = \frac{dx(t)}{dt} + x^2(t)$$ and I need to see if this system with input $x(t)$ and output $y(t)$ is time invariant and linear. I tried to break up the equation by substituting a linear operator D as the derivative.

So I get: $$y(t) = \frac{Dx(t)+x^2(t)}{D^2 + 1}$$

But the problem is, this equation is meaningless since you are dividing by the derivative operator. However, mathematically, it works out, so can I get $y(t)$ in this form and test for linearity or time invariance using this method?

Answer 1

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.