Linearity/Non-Linearity of $y(t) = x(t) +\cos(w_0 t)$

Question

I have the following System with (input x(t), output(y(t)), that I have to check for linearity and time-invariance.

$y(t) = x(t) + \cos(w_o t )$

I am able to show that it is time-variant. For linearity I am not that sure. Here is what I have done so far.

$$ y_1(t)= x_1(t) +\cos(w_0t)\\ y_2(t)= x_2(t) +\cos(w_0t) $$

Adding these give me $y_1(t) + y_2(t) = x_1(t) + \cos(w_0t) +x_2(t) +\cos(w_0t)$

Now $x_1(t) + x_2(t) \rightarrow $ system $\rightarrow x_1(t) + x_2(t) +\cos(w_0t)$.

Is this correct?

Answer 1

Doubled input != doubled output:

$$ 2x(t) \rightarrow \text{system} \rightarrow 2x(t) + \cos(\omega_0 t) \neq 2 y(t) $$

Not linear. You've also shown it: need $2\cos(\omega_0t)$ to satisfy $x_1(t) + x_2(t) \Leftrightarrow y_1(t) + y_2(t)$.