function [] = prob_3_16
% 
% 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.
% 
%-----

warning( 'I couldn''t find two initial conditions that gave two different solutions' ); 
  
solinit = bvpinit(linspace(0,5,10),@guess1);  
sol = bvp4c(@f,@bcs,solinit);

figure; 
plot( sol.x, sol.y(1,:), '-bo' ); hold on; 

solinit = bvpinit(linspace(0,5,10),@guess2);
sol = bvp4c(@f,@bcs,solinit);

plot( sol.x, sol.y(1,:), '-go' ); grid on;


function v = guess1(x)
% GUESS1 - 
%   
v =   [ x*(x-5); 1; pi/6 ];


function v = guess2(x)
% GUESS2 - 
%   
v  = [ 0.1*x*(x-5); 1; pi/6 ]; 


function [res] = bcs(ya,yb)
% BCS - 
%      
  
res = [ ya(1);
        ya(2)-0.5;
        yb(1);
      ];


function [ ode ] = f(x,y)
% F - 
% y(1)=y(t); y(2)=v(t); y(3)=phi(t);
% 
  
g = 0.032; 
nu = 0.02; 

ode = [  
    tan(y(3));
    -(g*sin(y(3)) + nu*(y(2)^2))/(y(2)*cos(y(3)));
    -g/(y(2)^2); 
      ];