A mass spring damper system is governed by x¨(t)+x(t)x˙(t)+x(t)=f(t)𝑥¨(𝑡)+𝑥(𝑡)𝑥˙(𝑡)+𝑥(𝑡)=𝑓(𝑡). This system is nonlinear linear time varying non-causal

To analyze the given mass spring damper system described by the equation $\ddot{x}(t) + x(t) \dot{x}(t) + x(t) = f(t)$, we need to identify the properties of this system:

1. Nonlinearity: The term $x(t) \dot{x}(t)$ indicates that the system is nonlinear. In linear systems, the principle of superposition holds, meaning the response caused by two inputs can be expressed as the sum of the responses of each input. The presence of products of the state variable $x(t)$ and its derivative $\dot{x}(t)$ violates this principle, confirming the nonlinearity.

2. Time Variation: The system is time-varying. To be time-invariant, the system's behavior must be the same regardless of when the input is applied. The term $f(t)$ suggests that the input function can change over time, which makes the system time-varying.

3. Causality: The system is causal since the output at any time $t$ depends only on the current and past values of the input $f(t)$ and the state $x(t)$ and its derivatives. There are no future values influencing the current output.

In summary, the mass spring damper system described by the equation $\ddot{x}(t) + x(t) \dot{x}(t) + x(t) = f(t)$ is a nonlinear, time-varying, and causal system.