The order of an ordinary differential equation is the number of primes in the highest order derivative
In this case, x is the highest order derivative, and so the order of the ODE is 3
The ODE in Problem 2 can be converted to a system of n first-order ODEs like this
First you get the highest-order derivative by itself on the left hand side of the equation
Then you define yourself some helper variables
If you have a second-order ODE, you just need to define yourself one helper variable
If I had a third-order ODE, Id have to define myself two helper variables
Theyd look something like x = v, v = a
But here, I just need one
Then the next step is to rewrite the first equation using the helper variable, like this
Now, it doesnt matter what you call your helper variable; you could choose whatever you want here
It is important to make sure that there are no derivatives on the right-hand side of these equal signs
Thats actually why we define the helper variables, so that we can rewrite everything on the right-hand side without any derivatives
So this set of equations is not the right answer, because this would be the set of equations that would correspond to a third-order ODE; thats quite different
This set of equations is almost right, except for this term right here
Remember, we want no derivatives on the right hand sides of the equal signs
This is the right answer
This one has a sign problem, and this one doesnt have the helper variable in it, so its not complete