The first task here is to plug the constants into the equations, to see what were working with

Here are the simple harmonic oscillator equations, and heres what I get if I plug in g = beta = 0; k = 2; and m = 0.5

Heres the initial condition were starting from

By the way, this transpose notation means Take a row vector and make it a column vector

It can also mean Take a column vector and make it a row vector

Now recall, forward Euler advances from the current state in the following fashion

It takes the place where you are, computes the slope at that point, and walks along that slope for one timestep to get the next point, delta t later

We have all this information except for the derivative vector

To get the values for that vector, we need to plug this into this

And if we do that  that is, we evaluate the derivative [x, v] when x = -1 and v = -2  we get this value

And then if we stuff that in here, we get the answer

And if you do the addition right, youll see that this one is the correct answer