Question

Section 8.1: Problem 3 Previous Problem Problem List Next Problem (1 point) Take the conservative equation X" + x2 - 64 = 0. Write down the corresponding first order system using the extra variable y: x = y = The critical points are on the x-axis at the following two x values (order them as they are on the number lin Compute the Jacobian matrix for all x and y: The behavior at the first critical point is The behavior at the second critical point is For behavior write one of: "saddle","source", "sink", "spiral sink", 'spiral source", "center"

Answer 1

xa x²64 = 0 » X"=64-22 9' = 64-27 For caulical points on x axis. (y=0) put y': 0, => 64-? = 0 = 2 = t8 : x = [78] & 1-8] : The Jacobian Matrix J. The lon) va (y) 7 Scanned with CamScanner

UND Ś JE TO 17 1-2x of At, 2=8. to 17 1-16 od. Let eigenvalue = a. (a)x( a) + 16 = 0 => 72--16 : => fi.4 So, this is centreo) At x = -8, Let A be eigenvalue. - 92-16=0 => 2=14 So, this is saddle Ics Scanned with CamScanner