Recently came across to solve system of linear differential equation with the matrix method, I have

Aryan Emery 2022-02-25 Answered
Recently came across to solve system of linear differential equation with the matrix method, I have learnt that the eigenvectors forms part of the complementary function.
A matrix method such as :
dydt=5x+7y
dxdt=9x+4y
which then we replace a vector x =(x,y) and a matrix M with entries (57,94)
Is there any reason for this (such as the one in ordinary Differential equation is due to the fact that the exponential is the eigenfunction if the differential which also exist as a matrix method, but why the eigenvectors in the front of it?)
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Expert Answer

e4mot1ic5bf
Answered 2022-02-26 Author has 6 answers
Make a variable substitution. If you have a homogeneous linear system of ODEs x˙=Ax, where A has a basis of eigenvectors v1,,vn and corresponding eigenvalues λ1,,λn, then make a substitution in terms of a new vector variable y, the coordinate vector of x with respect to v1,,vn. Then,
x=y1v1++ynvnx˙=y˙1v1++y˙nvn.
Hence, the system becomes,
x˙=Ax
y˙1v1++y˙nvn=A(y1v1++ynvn)
=y1Av1++ynAvn
=λ1y1v1++λnynvn.
By the linear independence of v1,,vn, we may equate the (non-constant, but still scalar) coefficients of vi to obtain
y˙i=λiyi,
for each i. In other words, we have now decoupled the system into n totally independent first order ODEs, in terms of the coordinates of y. These have the usual solutions:
yi=Aieλit,
for some constant Ai. Thus, changing back,
x=y1v1++ynvn=A1eλ1tv1++Aneλntvn.
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Katey Guzman
Answered 2022-02-27 Author has 4 answers

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