% example script using eulerSolve function

clc % clear screen
clear % clear all variables

xMin = 0;
xMax = 1;
yInitial = 1;
Cval = 2/yInitial + xMin*xMin;

% step size
h=0.1;
[xSol,ySol] = eulerSolve(xMin,yInitial,xMax,h);

% plot a graph.  
% To understand 'bo-' etc, see the MATLAB help
% (the b is blue, the o is the plot symbol, - is for a line)
hold off % make sure that this will be a new graph
plot( xSol,ySol, 'bo-', 'Linewidth', 3 )
% make the font nice and big 
% gca refers to properties of the current fig
set(gca,'FontSize',30); 

% compute exact solution 
nExact = 100;
for i=1:nExact
    xExact(i) = xMin + i*(xMax-xMin)/nExact;
    yExact(i) = 2/( Cval - xExact(i)*xExact(i) );
end

hold on  % add an extra line to the existing graph
% k stands for black here
plot( xExact,yExact, 'k-', 'Linewidth', 3 )
legend('h=0.1','exact','Location','NorthWest')

% smaller step size
h=0.02;
[xSol2,ySol2] = eulerSolve(xMin,yInitial,xMax,h);
% m is "magenta" here

hold off % NEW graph, created from scratch
% note: the order of these following commands does matter,
% we have to do the first "plot" before setting labels, fonts, etc
plot( xExact,yExact, 'k-', 'Linewidth', 4 )
xlabel('x')
ylabel('y(x)')
xlim([ xMin xMax]);
set(gca,'FontSize',30)  % need to do this again for the new graph
hold on %% add the numerical solutions
plot( xSol,ySol, 'bo-', 'Linewidth', 3 )
plot( xSol2,ySol2, 'mo-', 'Linewidth', 3 )
legend('exact', 'h=0.1', 'h=0.02', 'Location','NorthWest')
hold off

% this command will print to a file
print -dpdf 'eulerFig.pdf' -bestfit