function [] = prob_3_4()
% PROB_3_4 - 
% 
% 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.
% 
%-----

global d; 
d = 0.001; 

solinit = bvpinit(linspace(d,0.5,10),@guess);  
sol = bvp4c(@f,@bcs,solinit);
xall = linspace(0,0.5,100); yaall = sf(xall); 
p0=plot( xall, yaall, '-or' ); hold on; 
p1=plot( [ 0 sol.x], [ 0 sol.y(1,:)], '-bo' ); grid on;
axis( [ 0 0.5 0 0.5 ] );

function [ v ]  = guess(x)
% GUESS - 
%
y = x.*(1-x); 
yp = 1-2*x;

v = [ y; yp ];


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


function [ ode ] = f(x,y)
% F - 
% y(1)=y; y(2)=y'
% 

ode = [  y(2); 
	 -1/y(1) ];


function yout = sf(xall)
% SF - 
%   
yout = xall.*sqrt(-2*log(xall+eps));