a system begins at rest with the given values (3), the system has damped harmonic oscillator and damping constant provided by the equation (1), that is influenced by the eqn (2). find the equation of motion and find the complementary solution of x(t). find all the coefficients and show work please The image contains handwritten mathematical expressions related to a physics or engineering problem. Here's the transcription:1. \(\beta = 2\omega_0\) \hfill (1)2. \(\frac{d}{dt} F = F_0 \cos(\omega t)\) \hfill (2)3. \(x(0) = x_0 \quad \text{and} \quad \dot{x}(0) = 0\) \hfill (3)These equations are commonly seen in the context of differential equations, oscillatory motion, or systems involving harmonic motion. Equation (1) might relate to a parameter or constant defined in terms of angular frequency \(\omega_0\). Equation (2) involves a cosine function, suggesting a periodic or oscillating force \(F\), with amplitude \(F_0\) and angular frequency \(\omega\). Equation (3) provides initial conditions for a system, where \(x(0)\) is the initial position and \(\dot{x}(0)\) is the initial velocity.

Given: The three equations are given as follows: Introduction: In real oscillators, friction, or…

Answered: a system begins at rest with the given…