Solving DE Using Multiple Methods: Eigenvalues, Eigenvectors, and More

[encorelui2]

People I am fairly new with dealing with DEs (and here I was thinking I got it all )
Any who: I am working on some problems here and the professor wants us to use 2 methods to solve this DE. My issue is I can't figure out what the 2nd method is!
(dY)/dt=AY +F
$^{Y}$ = [7,4,4; -6,-4,-7; -2,-1,2] and $\textbf{}F$ = [-3;0;3]e$^{-iwt}$
Initial contions: $$Y(0)=\begin{pmatrix} 1\\ -2\\ 3\end{pmatrix},\; $$

I was able to find the eigenvalues & eigenvectors; hence the particular & general solutions. My issue i s I don't know of any other method to do this. Can anyone point me in the right direction?

Thanks

 

[Simon Bridge]

Is [7,4,4; -6,-4,-7; -2,-1,2] a matrix?

So you wrote:
$$Y=\begin{pmatrix} 7 & 4 & 4\\ -6 & -4 & -7\\ -2 & -1 & 2 \end{pmatrix},\; F=\begin{pmatrix} -3\\0\\-3 \end{pmatrix}e^{-i\omega t}$$ ... did I read that correcty?

Anyway, did you try writing out the three DEs and solving them individually?

 

[encorelui2]

Is [7,4,4; -6,-4,-7; -2,-1,2] a matrix?

So you wrote:
$$Y=\begin{pmatrix} 7 & 4 & 4\\ -6 & -4 & -7\\ -2 & -1 & 2 \end{pmatrix},\; F=\begin{pmatrix} -3\\0\\-3 \end{pmatrix}e^{-i\omega t}$$ ... did I read that correcty?

Anyway, did you try writing out the three DEs and solving them individually?

Yes u read that correctly. Wait! Are u saying to create something like this:

7y₁ + 4y₂ + 4y₃ - 3e^iωt
-6y₁ - 4y₂ - 7y₃
-2y₁ - y₂ +2y₃ + 3e^iωt

 

[Simon Bridge]

If Y is the matrix, as in post #1, then $\frac{d}{dt}Y = AF$ expands to: $$\frac{d}{dt}\begin{pmatrix} 7 & 4 & 4\\ -6 & -4 & -7\\ -2 & -1 & 2 \end{pmatrix} = A \begin{pmatrix} 7 & 4 & 4\\ -6 & -4 & -7\\ -2 & -1 & 2 \end{pmatrix}+\begin{pmatrix} -3\\0\\-3 \end{pmatrix}e^{-i\omega t}$$
... which doesn't make a lot of sense...

From your continuation, I suspect you mean:

$$\frac{d}{dt}\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix}= \begin{pmatrix} 7 & 4 & 4\\ -6 & -4 & -7\\ -2 & -1 & 2 \end{pmatrix}\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix} + \begin{pmatrix} -3\\0\\-3 \end{pmatrix}e^{-i\omega t}$$

My previous suggestion was anticipating a particular response ... naughty of me :)
What I mean is that you know more than one way of handling DEs ... I have to go, so I'll just leave you with these notes:
http://tutorial.math.lamar.edu/Classes/DE/SolutionsToSystems.aspx

Just look at the system different ways and see if there are ways to simplify it or make intellegent guesses ... maybe it will be susceptable to Laplace transforms?

 

[szynkasz]

It can be solved by successive substitutions:

$\begin{cases}y_1'=7y_1+4y_2+4y_3-3e^{i\omega t}\\y_2'=-6y_1-4y_2-7y_3\\y_3'=-2y_1-y_2+2y_3+3e^{i\omega t}\end{cases}\\\\<br /> \begin{cases}y_1''=7y_1'+4y_2'+4y_3'-3i\omega e^{i\omega t}\\y_2'=-6y_1-4y_2-\frac{7}{4}\left(y_1'-7y_1-4y_2+3e^{i\omega t}\right)\\y_3'=-2y_1-y_2+\frac{1}{2}\left(y_1'-7y_1-4y_2+3e^{i\omega t}\right)+3e^{i\omega t}\end{cases}$

Now we substitue for $y_3'$. Next we can eliminate $y_2$ in a similar way. Finally we get equation of 3rd order with $y_1$ only.

 

[HallsofIvy]

I would look for eigenvalues and eigenvectors of that coefficient matrix. I find that the characteristic equation is $r^3- 5r^2- 45r+ 153= (r- 3)(r^2- 2r- 51)= 0$. The eigenvalues are 3 and $1\pm\sqrt{51}$ all of which are real numbers.

An eigenvector corresponding to eigenvalue is (0, 1, 1).

I haven't tried to find the Eigen vectors corresponding to the other two eigenvalues. I suspect they are rather messy.