Homework 5 collaboration area
4.4 Problem 7. Can anyone provide some guidance on finding the Eigenvector for the imaginery eigenvalues in this problem? I am stuck trying to get the matrix to row reduce.
Re: 4.4 Problem 7: First you get the matrix that represents the differential equation: [[0 -2],[8 0]]. Solve the determinant for det(A - Lambda*I) = 0 = Lambda^2 + 16. Then solving for lambda, we obtain purely imaginary eigenvalues. When plugging in the eigenvalues, the trick to row reducing them (to get a bottom row of 0's) is to multiply by "i" and then divide out the constants (be careful to note that i^2 = -1). This should get you the eigenvectors.
4.4 Problem 11. Does anyone have advice as to how this problem should be approached? It isn't a system of equations, so how do we get the eigenvalues/vectors?
Reply: Say that y = y1, and y2 = y1'. Then, you will have a system.
4.5 Problem 1. What do we do about the y2 + y2^2? Specifically the square?
Reply: factor our the y2: y2(1+y2). Set the function equal to zero (to find the critical points).
So you should get two solutions(y2 = 0 and y2 = -1). To find the corresponding values of y1, set the second function (3 y1 = 0, and solve). The two critical points should come out to be (0,0) and (0,-1).
The next thing you'll need to do is linearize the system at the critical points. The problem is asking for the eigenvalues of the linearized system at the critical points and what they imply about the stability of the solutions near the critical points.
More on P. 150 # 7: Towards the end of Lecture 11 we were given the general solution of a linear system with complex eigenvalues, but Im a little confused as to why the book does not have c1 or c2 listed in the solution. Any thoughts? Thanks.
