1. An example of using euler.m
Problem 1.3.3
An example of using euler.m
to approximate the solution of an ODE for different stepsizes and compare with the exact solution

2. function [X, Y] = eul1_3(x,y,x1,n)
h = (x1-x)/n;
X = x;
Y = y;
for i = 1:n
k = f(x,y);
x = x + h;
y = y + h*k;
X = [X; x];
Y = [Y; y];
end
hold on
plot(X,Y, 'b-')
axis([ ])
X1 = linspace(0,0.5,21); Y1 = linspace(1,3, 21);
%draw direction field
% plot exact solution
yex = exact(X)
plot(X,yex,'g-')
% plot initial condition
y0 = 1; x0 = 0;
plot(x0,y0,'kp')
% define differential equation
function yp = f(x,y)
% warning: f must be vectorized, ie use x.^2
yp = x + 2*y;
% define exact solution
function ye = exact(x)
ye = .5*x + exp(2*x);

3. % This version solves problem 1.3.3 by Euler's method
% (blue line)
% and shows the exact solution (green line) for comparison
% To run: type
% [X, Y] = euler(0,1,0.4,4)
% [X, Y] = euler(0,1,0.4,8)
% [X, Y] = euler(0,1,0.4,16)
% these will plot the direction field (red lines), the exact solution
% (green line) and the approximate solutions for the three different
% step sizes (.1, .05, .025).
% Turn in the output for all cases; make a table of the values returned by the
% three different runs, for the x values: 0.1, 0.2, 0.3, 0.4 %
% x | y1 | y2 | y | yexact
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