Is a falling object equation of motion a non-linear differential equation?

Question

I'm attempting to model a falling object in state space form. Problem below.

I've modeled higher order differential equations as state space systems before but this problem is confusing me. The equation that's provided is clearly a differential equation, however, it includes a the independent variable itself (time) along with an initial condition.

I also think it's odd that I am being asked to model a second order system with three state variables (I believe two would suffice).

So my question, is a falling object equation of motion non-linear?

Answer 1

The equation that's provided is clearly a differential equation, however, it includes a the independent variable itself (time) along with an initial condition.

No, the equation that's provided is a solution to a differential equation. The differential equation is not provided.

I also think it's odd that I am being asked to model a second order system with three state variables (I believe two would suffice).

You're right, but it's still possible to do what the problem requests.

So my question, is a falling object equation of motion non-linear?

The differential equation you seek is linear.

Edit:

By the way, the problem is written incorrectly, which I only just noticed, because I expected to see the correct version! The equation given is $p=p_0+\dot pt +\frac12\ddot pt^2$, but it should be $p=p_0+v_0t +\frac12at^2$ where $v_0$ is the initial velocity and $a$ is the constant acceleration due to gravity. The equation given is actually wrong if evaluated at any time other than $t=0$, where it is trivially correct. It also happens that at $t=0$, we have $\dot p = v_0$, and it is always true that $\ddot p = a$, which is why the author(s) made the mistake of conflating these expressions.