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Is the book incorrect here? Systems of diff eqs.

  1. Nov 19, 2012 #1

    Zondrina

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    1. The problem statement, all variables and given/known data

    Here is the question from my book along with their solution to the problem :

    Part 1 : http://gyazo.com/6467751155edcefec6cc583d164d2ae7
    Part 2 : http://gyazo.com/d91752f0a18a72af96f6afc74117f011
    Part 3 : http://gyazo.com/7a17dbca3e414d0ddc8458abce2eda9c

    2. Relevant equations

    I'll denote the coefficient matrix by the letter A.
    I denote my null space operator by N.
    t is the transpose of a vector.

    3. The attempt at a solution

    Now I'm almost positive the book is wrong here. Here's my solution.

    Assume that [itex]x = εe^{λt}[/itex] so our equation becomes :

    [itex]λεe^{λt} = Aεe^{λt} \Rightarrow (A - λI)ε = 0[/itex]

    So we seek eigenvector(s) ε such that the above equation is satisfied. So lets find the eigenvalues for A ( I'll skip some of the boring algebra here ) :

    [itex]det(A - λI) = (λ-3)(λ+1)[/itex] Thus λ1 = 3 and λ2 = -1 are eigenvalues for A.

    Now to find the eigenvectors for the eigenvalues, we seek the null space of A - λjI. So :

    [itex]ε(λ_1) = N(A - 3I) = span\left\{{[1/2 \space \space 1]^t}\right\} [/itex]

    [itex]ε(λ_2) = N(A + I) = span\left\{{[-1/2 \space \space 1]^t}\right\} [/itex]

    See ^ for some reason I get different eigenvectors than the book does. Everything else that follows after this is exactly the same as the book except my eigenvectors.

    Have I made an error somewhere? Or is the book wrong in this case? Some clarification on this would be greatly appreciated.

    Thanks in advance.
     
    Last edited: Nov 19, 2012
  2. jcsd
  3. Nov 19, 2012 #2

    Ray Vickson

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    No, you are getting the same eigenvectors, but just scaled differently. Instead of <1/2,1> the book uses <1,2>, and instead of <-1/2,1> the book uses <1,-2>. Eigenvectors are, of course, not unique: any nonzero scalar multiple of an eigenvector is also an eigevector for the same eigenvalue.

    RGV
     
  4. Nov 19, 2012 #3

    Zondrina

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    Ahh yes, I remember now. So multiply my first one by 2 and my second one by -2 to get the same answer as they have.

    Thanks for clearing that up for me RGV.
     
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