<p> Mat 275 Modern Differential Equations Dr. Firozzaman Department of Mathematics and Statistics Arizona State University Lab 2 due on August 1, 2006</p><p>Examples: Try all (Do not print, you need to print solutions of the homework problems only) Files containing MATLAB commands are called m-files and have a .m extension. They are two types:</p><p> a) A script is simply a collection of MATLAB commands gathered in a single file. The value of the data created in a script is still available in the command window after execution. </p><p>Example 1. Script file save as plot1.m (type as a file and save)</p><p>Plotting a function y  22e x / 5  5x  25 using window [0, 5]x[-5, 10]</p><p> x = 0:1:5; y = 22*exp(x./5)-5*x-25; plot(x,y,’r-’, ’LineWidth’,2); axis([0,5,-5,10]); grid on; title(’f(x)=(22 e^(x/5)-5x-25)’); xlabel(’x’); ylabel(’y’); Then run your program and get the plot on MATLAB prompt by typing >>plot1</p><p> b) A function is similar to a script, but can accept and return arguments. Unless otherwise specified any variable inside in a function is local to the function and not available in the workspace. A function invariably starts with the command function output = function_name(input)</p><p>Example 2. Script + Function (two separate files)</p><p> i) Save the file as function1.m</p><p> function y=function1(x) y = 22*exp(x./5)-5*x-25;</p><p> ii) Save the file as plot2.m x = 0:1:5; y = feval(@function1,x); plot(x,y,’r-’, ’LineWidth’,2); axis([0,5,-5,10]); grid on; title(’f(x)=(22 e^(x/5)-5x-25)’); xlabel(’x’); ylabel(’y’); Then run your program and get the plot on MATLAB prompt by typing >>plot2 c) Script + Function (one file) save as plot3.m</p><p> function plot3 x = 0:1:5; y = feval(@function1,x); plot(x,y,’r-’, ’LineWidth’,2); axis([0,5,-5,10]); grid on; title(’f(x)=(22 e^(x/5)-5x-25)’); xlabel(’x’); ylabel(’y’); %------function y=function1(x) y = 22*exp(x./5)-5*x-25;</p><p>Then run your program and get the plot on MATLAB prompt by typing >>plot3</p><p> d) Numerical solution by Euler Method</p><p>As an example consider the initial value problem (IVP)</p><p> dy  2y, y(0)  3 dt Follow the steps:</p><p>>>clear t y >>y(1)=3; t(1)=0; h =0.1; >>f=inline(’2*y’, ’t ’, ’y ’); >>y(2)=y(1)+h*f(t(1),y(1)), t(2)=t(1)+h, >>y(3)=y(2)+h*f(t(2),y(2)), t(3)=t(2)+h, >>y(4)=y(3)+h*f(t(3),y(3)), t(4)=t(3)+h, >>y(5)=y(4)+h*f(t(4),y(4)), t(5)=t(4)+h, >>y(6)=y(5)+h*f(t(5),y(5)), t(6)=t(5)+h, >>[t(:),y(:)] >>plot(t,y);axis tight</p><p>------Homework: You need to print your work and turn in.</p><p>1. Write an m-file to plot the function y  3e2x with suitable x and y values dy 2. Use Euler Method to plot the solution curve of the IVP  t  y, y(0)  1 dt 3. Find eigenvalues and eigenvectors of A = magic(3) (Optional) ------</p><p>Try the following matrix operations. Do not print. >> A=[8, 1,6; 3, 5, 7; 4, 9, 2] >> row1=[8, 1, 6]; row2 =[3, 5, 7]; row3 =[4, 9, 2]; A=[row1; row2; row3] Also try using columns like co1, col2, col3</p><p>Try the following and understand what you get back:</p><p>>>A(2, 3) % coefficient of A in 2nd row , 3rd column >>A(1,:) % first row of A >>A(:,3) % 3rd column of A >>I=eye(3) >>B=magic(3) %can you find what is magic of this matrix? >>C= A+B >>D = A*B >>[S,D]= eig(A) </p><p>%determines the eigenvectors of A (columns of S) and associated eigenvalues (diagonal coefficients of D) note that the eig function has one input and two outputs arguments. </p><p>You may verify eigenvalues and eigenvectors of the problem we have solved in the class (Section 5.2) ------HAVE FUN ------</p>