(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

solve the system of first-order linear differential equations:

(y1)' = (y1) - 4(y2)

(y2)' = 2(y2)

using the equation:

(λI -A)x = 0

2. Relevant equations

using eigenvectors and eigenvalues

in the book 'Elementary Linear Algebra' by Larson and Falvo - Section 7.4 #19

3. The attempt at a solution

(y1)' = (y1) - 4(y2)

(y2)' = 2(y2)

makes the matrix:

[1 -4]

[0 2]

(λI-A)x = 0

[λ-1 4]

[0 λ-2]

which gives the eiganvalues:

(λ-1)(λ-2)

λ = 1

λ = 2

then input it back into the original matrix A:

λ = 1

(1I - A) =

[0 4]

[0 -1]

which reduces to:

[0 0]

[0 1]

and creates the eiganvector:

[0]

[1]

then continued onto:

λ = 2

(2I - A) =

[1 4]

[0 0]

which creates the eigenvector:

[1]

[-4]

so then I put the two eigenvectors together and create P:

[1 1]

[0 -4]

then I need to find P^-1 so that I may use PAP^-1 to find the differential equation:

I set P equal to the identity matrix and solve (I think this is where things start to go wrong...)

[1 1/4]

[0 -1/4]

then I solve for PAP^-1

[1 1] [1 -4] [1 1/4]

[0 -4] [0 2] [0 -1/4]

I take AP^-1 first in the multiplication and get:

[1 5/4]

[0 1/2]

and then add on P and get:

[1 7/4]

[0 2]

and that just looks wrong, I stopped there in fear of continuing further, but the answer IN THE BOOK reads:

(y1) = (C1)e^t - 4(C2)e^2t

(y2) = (C2)e^2t

and it looks like if I were to continue it would not even come close to that answer...Where did I go wrong?

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# Solve the system of linear differential equations using eigenvectors and eigenvalues

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