Problem 1.3.3

1. 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([0 0.5 1 3]) X1 = linspace(0,0.5,21); Y1 = linspace(1,3, 21); %draw direction field df(@f,X1,Y1,'x','y','dirfield','r',0); % 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 = 0.5 - 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 | y3 | yexact % --------------------------------------------------- % 0.1 | 1.2500 | 1.2600 | 1.2655 | 1.2714 % 0.2 | 1.5400 | 1.5641 | 1.5775 | 1.5918 % 0.3 | 1.8780 | 1.9216 | 1.9459 | 1.9721 % 0.4 | 2.2736 | 2.3436 | 2.3829 | 2.4255 % ---------------------------------------------------