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

In aerodynamics, one encounters the following initial value problem for Airy’s equations:

y''(x) + xy = 0, y(0) = 1, y'(0) = 0

Using the Runge-Kutta method with h=0.005 and determine values between x=0 and x=10 sufficient to sketch the relationship.

2. Relevant equations

y''(x) + xy = 0, y(0) = 1, y'(0) = 0

I think,

k1 = h*f(Xn, Yn)

k2 = h*f(Xn+h/2, Yn+k1/2)

k3 = h*f(Xn+h/2, Yn+k2/2)

k4 = h*f(Xn + h, Yn + k3)

3. The attempt at a solution

From what I have read you cant do second order ODE using runge kutta without breaking it into a system of first order ODEs so thats what I tried.

I tried:

d2y/dx2 + xy = 0

dy/dx = z, y(0) = 1

dz/dx + xy = 0

dz/dx = -xy, z(0) = 0

I dont know if that is right or not and if it is I have no idea where to go from here.

Thanks for any help.