Free Damped Vibration

Question

I have basic ideas about superposition principle and linear systems especially in algebraic equations, and wave superposition as far as response is concerned. But i really can not see the same thing, intuition or understanding in the following image.

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Answer 1

It basically asserts that if you can find two solution to the equation, then all linear combinations of these solutions are also a solution. To see that assume $x(t)$ and $y(t)$ are two solutions to the differential equation. Then $\alpha x(t) + \beta y(t)$ is also a solution, because $$\begin{align} \frac{d^2(\alpha x + \beta y)}{dt^2} + 2\zeta \omega_n \frac{d(\alpha x + \beta y)}{dt} + \omega_n^2 (\alpha x + \beta y) &= \\ \alpha \ddot{x} + \beta \ddot{y} + 2\zeta \omega_n \alpha \dot{x} + 2\zeta \omega_n \beta \dot{y} + \omega_n^2 \alpha x + \omega_n^2 \beta y &= \\ \alpha (\ddot{x} + 2\zeta \omega_n \dot{x} + \omega_n^2 x) + \beta (\ddot{y} + 2\zeta \omega_n \dot{y} + \omega_n^2 y) &= 0 \end{align}$$