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 February 13th, 2011, 07:25 PM #1 Member   Joined: Feb 2011 Posts: 40 Thanks: 0 Help with differential systems dealing with linearity and li 1) Verify the linearity principle for linear, nonautonomous systems of differential equations? 2) Consider all linear systems with two zero eigenvalues. which of these systems are conjugate? prove this. 3) Prove that a linear map T : R2 -> R2 is a homeomorphism if and only if it is nondegenerate? I'm trying to figure these questions out for practice for an upcoming midterm. I don't know where to start or how to start any of them I know that nondegenerate just means that the det =\= 0 and that the linearity principle is "if X' = AX is a planar linear system for which Y_1(t) and Y_2(t) are both solutions, then the function aY_1(t)+bY_2(t) is also a solution to this system."
 February 14th, 2011, 01:27 PM #3 Member   Joined: Feb 2011 Posts: 40 Thanks: 0 Re: Help with differential systems dealing with linearity an Thanks for the reply. I actually ended up doing what you mentioned for number 1 and number 3. As for the definition of the kind of conjugate that I was referring to, in my book it has: Given x'=ax and x'=bx have flows phi_a and phi_b. They are conjugate iff there exists a homeomorphism h: R^2 -> R^2 that satisfies phi_b(t, h(X_0))=h(phi_a(t,X_0))
 February 17th, 2011, 04:35 PM #4 Senior Member   Joined: Jun 2010 Posts: 618 Thanks: 0 Re: Help with differential systems dealing with linearity an Hi again. I forgot about your problem for a while ... did you end up figuring out the last bit about conjugation? Let me know how the proof turned out, or if you need any more help on it. -Ormkärr-

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### Consider all linear systems with two zero eigenvalues. Which of these systems are conjugate? Prove this.

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