Second Order Runge Kutta for Simple Harmonic Motion

[Abigail1997]

Homework Statement

The ordinary differential equation describing shm is
d^2x/dt^2=-w^2x
where x is the displacement, t is the time and w is the frequency. If x=0 at t=0, the analytical solution is x=Asin(wt), where A is the amplitude.

1) Rewite equation 1 as two first oder ode's suitable for solution using Runge Kutta Methods
2)Determine the second order runge-kutta solution for this system after the first time step h and show the leading error term in x(h) is proportional to h^3

Homework Equations

k1=hf(xn,yn)
k2=hf(x+h, y+k1)
y_(n+1)=y_n+(1/2)k_1+(1/2)k_2

The Attempt at a Solution

I have completed part 1 and got dx/dt=v and dv/dt=-w^2x but I am unsure how to proceed. The lecturer didn't do a great job of explaining the method and I don't know how to do it when you have two equations and are not given the step size.

 

[DrClaude]

The relate what you got and the equations of the Runge-Kutta algorithm, set $y_1 \equiv x$ and $y_2 \equiv v$, and remember that
$$
\frac{d y_n}{dt} = f_n
$$