function [t,x] = rk4(f, tRange, x0, n)
% rk4: 4th order Runge-Kutta method for solving a single ODE
% f: function f(t,x) defining x'
% tRange: [t0,tEnd], the starting and ending times
% x0: initial value of x
% n: number of time steps so time values are t(1) to t(n+1)
% [t,x]: an m by 2 matrix containing the (t,x) values
% t and x are each column vectors.

t0 = tRange(1);
tEnd = tRange(2);
h = (tEnd - t0) / n;
t = (t0 : h : tEnd)'; % column vector
x = zeros(length(t),1);

x(1) = x0;
for k = 1 : length(t)-1
   k1 = h*f(t(k),x(k));
   k2 = h*f(t(k) + h/2, x(k) + k1/2);
   k3 = h*f(t(k) + h/2, x(k) + k2/2);
   k4 = h*f(t(k) + h,   x(k) + k3);
   x(k+1) = x(k) + (k1 + 2*(k2 + k3) + k4) / 6.0;
end