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Coupled Differential Equations

  1. Jan 26, 2016 #1
    1. The problem statement, all variables and given/known data

    Hi. I am trying to solve a problem on renormalisation group flow and have come across the following coupled equations that I need to solve:

    Λ ∂g/∂Λ = b.m

    Λ ∂m/∂Λ = -2.m + a.g

    Where a and b are just constants. I need to find g(Λ) and m(Λ).

    2. Relevant equations

    I thought this could perhaps be solved by turning it into a matrix equation and then diagonalising and expressing g(Λ) and m(Λ) as sums of eigenvectors.


    3. The attempt at a solution

    First I found the eigenvalues of the matrix to give:

    λ+ = -1 + √(1+ab)
    λ- = -1 - √(1+ab)

    And then I found the eigenvectors of the operator Λ ∂/∂Λ :

    ν+(Λ) = (Λ/μ)λ+ ν+(μ)

    Where μ is just some integration constant. A similar expression holds for ν-

    I then decomposed:

    g(Λ) = α ν+ + β ν-

    m(Λ) = γ ν+ + δ ν-

    But I don't really know where to go from here? Any help would be greatly appreciated!
     
  2. jcsd
  3. Jan 26, 2016 #2

    Ray Vickson

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    Homework Helper

    Try a solution of the form
    [tex] g(x) = A_1 x^r + B_1 x^s, \: m(x) = A_2 x^r + B_2 x^s [/tex]
    where the ##A_i, B_i, r, s## are constants (and I write ##x## instead of ##\Lambda##).
     
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