I am trying to solve this differential equation system:
I'm trying to write this into a matrix in the form of:
x' = A x
Where "A" is my matrix for this equation system. But when I try to create the matrix out of my own logic thinking, I get this:
This doesn't make any sense though since the determinant here is 0, which it shouldn't be. So what am I doing wrong?
Why do you say that does not make sense? Why can the determinant not be 0? That matrix has eigenvalues 0 (because the determinant is 0), i and -i. That means eachwill be of the form A+ B cos(x)+ C sin(x).
But, personally, I wouldn't use a matrix solution. Differentiating the first equation again, we haveso that
. That equation has general solution
. Then
so that
. Finally,
so that
just as I said.
How did you conclude that the matrix has the eigenvalues i and -i?Originally Posted by HallsofIvy
Why do you say that does not make sense? Why can the determinant not be 0? That matrix has eigenvalues 0 (because the determinant is 0), i and -i. That means each
But, personally, I wouldn't use a matrix solution. Differentiating the first equation again, we have
My assignment says I need to find the complete complex solution for the differential equation system. And that the facit is this:
By looking at the facit, it says that the eigenvalues are 0, i and -i but our textbook didn't mention anything about a matrix with eigenvalues of 0 so I have no idea how to conclude this.
We haven't been taught any other way of solving this. According to my textbook, once I have the eigenvalues and eigenvectors, I can insert them into this solution:
where "lambda" is my eigenvalue and "v" is my eigenvectors. Since I have a system of 3 equations, I should end up with 3 solutions.
But nobody ever mentioned to me that a matrix with determinant 0 also has the eigenvalues i and -i?
I understand that -i is the conjugated value of i.
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