function RungeKuttaExample()
    %this function solves the ode dQ/dt = cos(t) - lambda*Q for the initial condition
    %Q = (lambda*cos(t)+sin(t))/(1+lambda^2) at t=0 using a fourth order Runge-Kutta scheme.

    lambda = 2;

    %timestep to be used
    t_step = 12*pi/300;

    %number of timesteps
    n_steps = 301;

    %initalise a vector which is the required length
    Q = zeros(1, n_steps);

    %set the initial condition
    Q(1) = lambda/(1+lambda^2);

    %start at time = 0
    t = 0;

    %step through each of the timesteps
    for i = 2:n_steps
        k1 = t_step*func(t, Q(i-1), lambda);
        k2 = t_step*func(t + t_step/2, Q(i-1) + k1/2, lambda);
        k3 = t_step*func(t + t_step/2, Q(i-1) + k2/2, lambda);
        k4 = t_step*func(t + t_step, Q(i-1) + k3, lambda);

        %calculate the next step
        Q(i) = Q(i-1) + (k1+2*k2+2*k3+k4)/6;

        %move onto next timestep ready for the next loop
        t = t + t_step;
    end

    %set the time axis
    t_axis = 0:t_step:(12*pi);
    %plot numerical output
    plot(t_axis, Q, 'k')

    %plot analytical solution
    %set the time axis
    t_axis = 0:0.01:(12*pi);
    Q_real = (lambda*cos(t_axis)+sin(t_axis))/(1+lambda^2);
    plot(t_axis, Q_real, 'm*')


end

function [out] = func(t, Q, lambda)
    %this function evaluates the right hand side of the governing equation.
    out = cos(t) - lambda*Q;
end