Math 293, Homework 8 due in lecture on Wednesday October 23, 1996. (*) = don't hand in *) You should be able to do problems of the form 5.2.1-37 1) 5.2.13. Pick and state some initial condition and find c1 and c2 to satisfy that initial condition. 2) Let x(t) = [ x1(t) x2(t) x3(t) x4(t) ]' where []' is the transpose of []. A = [ -10 -2 3 -4; 5 -6 7 -8; 1 -2 3 5; e -pi 1 -1 ] ; ( where pi=3.1415... and ln(e) = 1 ) You are given that x satisfies the system of ODEs x' = A x and the initial condition x(0) = [ 0 0 1 -3 ]' a) Find x(2) using the eigenvalue and eigenvector method. That is find the 4 numbers x1(2), x2(2), x3(2), and x4(2). You may assume, correctly, that A has 4 eigenvectors. (The course WWW page has enough samples to get you through the needed MATLAB commands. The text and lecture indicate the recipe that is to be applied.) Hand in a tidy script file, not a rambling interactive session of commands.) b) Find x(2) using ODE23 or ODE45. (If any of your 4 answers from parts (a) and (b) differ by more than a percent you have made a mistake.) c) Optional challenge 1: How close can you get the answers of (a) and (b)? At that accuracy which takes less computer time to find? d) Optional challenge 2: How compact can you make your solutions?