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