Looking for examples on differential equations. please write detailed steps. for what I ask

you can choose any equation of your liking to make examples with what is asked below please be as detailed as you can

I want to learn and be able to figure out how to do it. your help is much appreciated

-write the ODE of the model of the system (rate in - rate out)

-write the direction field (6x6) explain how

-find the solution for the initial value problem

Given a differential equation (any)

-find the direction field

-use appropriate theorem to prove a unique solution exists (Linear and Non Linear for both please)

-use euler's method and find approximate solution

-solve initial value problem (IVP)

-find existence and uniqueness by numerical approach and direct approach (use solution method)

-sketch solution and piece wise approximation

-which one is larger the approximation or solution and why?