Coupled Differential Equations

[dman12]

Homework Statement

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).

Homework Equations

[/B]
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!

 

[Ray Vickson]

Homework Statement

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).

Homework Equations

[/B]
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!

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