# Second Order Runge Kutta for Simple Harmonic Motion

1. Sep 22, 2016

### Abigail1997

1. The problem statement, all variables and given/known data
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

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

3. 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.

2. Sep 22, 2016

### Staff: Mentor

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
$$\frac{d y_n}{dt} = f_n$$