Show that decomposition does not hold for non-linear system

Question

The solution to an inhomogeneous differential equation can be split up into homogeneous solution and a particular solution (forced response).

Another way to split up the solution to an inhomogeneous differential equation is in a zero-input response and a zero-state response.

The zero-input response is the system's response to its own internal initial conditions - no input signal is applied. The zero-input response is the homogeneous solution to the system's differential equation, using the initial conditions at $t=0^-$.

The zero-state response is the system's response to only the input signal - all initial conditions set to $0$. The zero-state response is found by convolving the system's impulse response $h(t)$ with the input signal $x(t)$.

This excerpt is from Lathi's signal processing and linear systems:

Lathi claims that for a linear system, one can show that the decomposition property holds.

Question:

How can I show that the decomposition property for a non-linear system, for example $\dot{y}(t) + y(t) = x(t) + 1$, does not hold?

Answer 1

Assume that $y_0(t)$ is the zero-input response. Then $y_0(t)$ must satisfy

$$\dot{y}_0(t)+y_0(t)=1\tag{1}$$

because $x(t)=0$.

Now let $y_1(t)$ be the zero-state response to an input $x(t)$, satisfying

$$\dot{y}_1(t)+y_1(t)=x(t)+1\tag{2}$$

If the decomposition property holds, the function $y_2(t)=y_0(t)+y_1(t)$ must be the response to $x(t)$ with possibly non-zero initial conditions. I.e., $y_2(t)$ should satisfy

$$\dot{y}_2(t)+y_2(t)=x(t)+1\tag{3}$$

However, adding Equations $(2)$ and $(3)$ gives

$$\dot{y}_2(t)+y_2(t)=x(t)+2\tag{3}$$

which shows that for the given system the decomposition property doesn't hold.