Problem involving second order DFQ

I am looking for a detailed solution. I need set up, solving, and final result. Therefore, I need solution of the second order DFQ. Finally for the description of the movement as time goes to infinity, I need what type of motion is that. a. An 8lb weight is attached to a spring suspended from the ceiling. When the weight comes to rest at equilibrium, the spring has been... Please see attached.

Initial value problem

Solve the IVP... Please see attached.

Existance and uniqueness of solution

I just need to know in a simple statement for each problem if the solution exists, if it is unique, and what is the interval. Based on what is unique?

First order DFQ W/Initial Value

I am looking ofr the solution of this problem. It involves a little bit of theory in the second part. In the solution give a detailed solution showing all assuptions and theorems.

Explicit Euler Method

Was Euler the ancient fortune-teller? He almost was. One of his principles, the explicit method says that your future days could be predicted from your present day knowing your past provided your time frame is not too large. Have a look at the solution to a system of non-linear differential equations system using explicit or forward Euler method. See attached file.