how to use eigenvalues and eigenvectors to find general solutions to a DE
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For a quick refresher on calculation of eigenvalue/eigenfunctions, go to:
http://mathworld.wolfram.com/Eigenvalue.html
Please share your work with us ...
If you are stuck at the beginning tell us and we'll start with the definitions.
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HallsofIvy
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You understand, do you not, that you can write the system of equations as
\(\displaystyle \begin{bmatrix}\frac{dx}{dt} \\ \frac{dy}{dt}\end{bmatrix}= \begin{bmatrix}1 & 3 \\ 2 & 2 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}\)
There are several different ways to solve such an equation and since you have not shown any attempt of your own, we don't know which methods you have been taught. This is what I would do:
If P is the matrix with columns being the Eigenvectors of matrix A, then \(\displaystyle P^{-1}AP\) is the diagonal matrix, D, having the eigenvalues of A on its diagonal
So if we have the matrix differential equation \(\displaystyle \frac{dX}{dt}= AX\), we can multiply both sides by \(\displaystyle P^{-1}\) to get \(\displaystyle \frac{d(P^{-1}X)}{dt}= P^{-1}Ax\) and we can write that as \(\displaystyle \frac{d(P^{-1}X)}{dt}= P^{-1}A(PP^{-1})X= (P^{-1}AP)(P^{-1}X)\)
Let \(\displaystyle Y= P^{-1}X\) and we can write the equation as \(\displaystyle \frac{dY}{dt}= DY\) which, because D is diagonal will give two separate, uncoupled, equations for the components of Y. Then, of course, \(\displaystyle X= PY\).
\(\displaystyle \begin{bmatrix}\frac{dx}{dt} \\ \frac{dy}{dt}\end{bmatrix}= \begin{bmatrix}1 & 3 \\ 2 & 2 \end{bmatrix}\begin{bmatrix}x \\ y \end{bmatrix}\)
There are several different ways to solve such an equation and since you have not shown any attempt of your own, we don't know which methods you have been taught. This is what I would do:
If P is the matrix with columns being the Eigenvectors of matrix A, then \(\displaystyle P^{-1}AP\) is the diagonal matrix, D, having the eigenvalues of A on its diagonal
So if we have the matrix differential equation \(\displaystyle \frac{dX}{dt}= AX\), we can multiply both sides by \(\displaystyle P^{-1}\) to get \(\displaystyle \frac{d(P^{-1}X)}{dt}= P^{-1}Ax\) and we can write that as \(\displaystyle \frac{d(P^{-1}X)}{dt}= P^{-1}A(PP^{-1})X= (P^{-1}AP)(P^{-1}X)\)
Let \(\displaystyle Y= P^{-1}X\) and we can write the equation as \(\displaystyle \frac{dY}{dt}= DY\) which, because D is diagonal will give two separate, uncoupled, equations for the components of Y. Then, of course, \(\displaystyle X= PY\).
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how to find general solution of differential equations
ive got 4 and -1 for eigenvalues
[1,1] and [1,-2/3] for eignvectors
not quit sure if i got the second eigenvector right
and i got linear independence for question b by finding out whether determinant is 0 or not and I'm also not sure if
thats right as the second eigenvector is not that clear
and now im stuck on question c please help
ive got 4 and -1 for eigenvalues
[1,1] and [1,-2/3] for eignvectors
not quit sure if i got the second eigenvector right
and i got linear independence for question b by finding out whether determinant is 0 or not and I'm also not sure if
thats right as the second eigenvector is not that clear
and now im stuck on question c please help
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HallsofIvy
Elite Member
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- Jan 27, 2012
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If you do not understand my reply above, what questions do you have?
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