function rk

% Numerical solutions and plots using Runge-Kutta implementation
% This command sets options for the ode routine below.
% for other options, type "help ode45",

options = odeset ('RelTol', 1.0e-06);

% We use a loop here to get solutions for different initial data.
% in this case, x0 ranges from -2 to 2 in increments of .1
% YOU WILL NEED TO CHANGE THIS FOR EACH PROBLEM, OR YOUR PLOTS
% MIGHT LOOK VERY STRANGE!

x0small = -2;
x0big = 2;

for x0 = [x0small:0.1:x0big],

% The next command invokes the fourth order runge-kutta scheme for the
% ODE x' = odefunction.  [0 5] specifies the time interval
% over which it is solved, x0 is the initial data,
% and "options" are specified above.
% notice that we never set the timestep size: this is because
% ode45 does it for us!

  [t,x] = ode45( @odefunction, [0 5], x0, options);

  % 't' is a vector of all t values
  % and the 'x' is a vector of all corresponding x values.
  % Therefore to plot the solution, we do the following:

  plot(t,x); hold on;  %'hold' keeps the current plot visible next time 'plot' is used

  % look at the help menu or type 'help plot' to see many more
  % graphical features;  for example, 'plot(t,x,'.') will plot points
  % instead of lines.

end;  %end the loop over initial conditions

%now were done plotting, so........
xlabel('t'); % label axes 
ylabel('x');
axis([0 5 x0small x0big]);  % only show the "relevant" part of the plot
hold off;  %this turns the 'hold' feature off

%-------------------------------------------------------------
% This defines the function f(x) in the equation dx/dt = f(x);
% you will edit this for each problem

function f = odefunction(t,x)

f = -x^3 + x;