Check this out.
I have been trying to solve this equation using matrix methods to find the general solution for x1(t) and x2(t)
(There should be a dot above the x1, x2 to show first order)
x1 = 4x1+ 7x2
x2 = -6x1-9x2
Now i have been shown how to do a second order question so i have used a similar method with this which gets me to this point.
det (a-lambda^2I) = (4-lambda^2)(-9-lambda^2)+42
(lambda^2+2)(lambda^2+3)
lambda^2 = -2 and lambda^2=-3
This is where its different to the question i have been through
As this would mean lambda is j1.414 and j1.732
The second order question i have been through comes out with lambda +or- 2j and +or- j which is then obvious how to continue.
Could somone point me in the right direction or to where ive gone wrong?
I have an exam on wednesday and realy need to be able to work with any combination of question they give me, It could be first or second order.
I didnt think it mattered which kind i was until differentiating the general solution part.
Thank you!
You're setting up the wrong characteristic equation. Your system looks like this:
where
The solution to the system is
We now have to make sense of the exponential. You use the series definition:
It's the usual series expansion of the exponential function. We can easily find the nth power of if we can find an invertible such that where is diagonal. For then
Taking the nth power of a diagonal matrix is the same as taking the nth power of the diagonal elements, component-wise. Solving is the diagonalization problem. To diagonalize, assuming it's possible to do this, you find the eigenvalues and eigenvectors. The eigenvalues make up the diagonal of , and the eigenvectors make up the columns of . Hence the importance of the eigenvalues for a system of ODE's.
Now, to find the eigenvalues, you have to set not That's where you first went wrong. So carry that correction through, and see what you come up with.
Ok ive done it using dec (A-lambdaI)=0 which would give me Lambda= -2 and -3
Then putting that back into the matrix i get.
For lambda -2 6x1=-7x2
Lambda -3 x1=-x2
x1 = 6(Ae^-2jt + Be^-3jt)
x2 = -7(Ae^-2jt -Be^-3jt)
to be honest im lost, i do not understand the exponential part.
I dont understand why there is a 1 in the first vector bracket? i only get either 6 and -7 or 7 and -6.
Then i realy do not understand where the final solution comes from I can see that D and e^Dt are from the two eigen values so do you then work the determinant for that out?
Please forgive my lack of understanding but i realy am trying to understand.
Any nonzero scalar multiple of an eigenvector is an eigenvector. Why is that, might you ask? Well, recall that a vector is an eigenvector of the matrix if and only if and for some scalar which is called the eigenvalue corresponding to . So, suppose that is an eigenvector of the matrix with corresponding eigenvalue . Let be a nonzero scalar. Then Since and (since and ), it follows by definition that is an eigenvector of matrix .I dont understand why there is a 1 in the first vector bracket? i only get either 6 and -7 or 7 and -6.
Now, if you look at my and multiply by , you might get a more recognizable eigenvector. Does that make sense?
I don't compute the determinant, I just compute the exponentialI can see that D and e^Dt are from the two eigenvalues; so do you then work the determinant for that out?
I'm not sure I understand what you're not understanding. The solution to this problem: pepper me with questions. Anything at all you don't understand, ask about. Ok?