function [] = prob_3_18
% 
% Written by:
% -- 
% John L. Weatherwax                2005-04-13
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

L = 10; 
PoverR = 0.001; 

guess = [ 1; 0; 1/sqrt(2); 1/sqrt(2) ];
solinit = bvpinit(linspace(0,L,10),guess);  
sol = bvp4c(@f,@bcs,solinit,[],PoverR);


figure; 
%plot( sol.y(3,:), sol.y(4,:), '-bo' ); hold on; grid on;
xint = linspace(0,L);
Sxint = deval(sol,xint);
plot( Sxint(3,:), Sxint(4,:), '-bo' ); hold on; grid on;



function [res] = bcs(ya,yb,PoverR)
% BCS - 
%      

res = [ ya(1);
        yb(2);
        ya(3);
        ya(4)-1;
      ];


function [ ode ] = f(x,y,PoverR)
% F - 
% y(1)=phi(t); y(2)=phi'(t); y(3)=x(t); y(4)=y(t)
% 

ode = [  
    y(2);
    -(PoverR)*cos(y(1));
    +cos(y(1));
    -sin(y(1));
      ];