# Methods of solving DE

1. Jul 4, 2013

### 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

Last edited: Jul 4, 2013
2. Jul 4, 2013

### 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?

3. Jul 4, 2013

### encorelui2

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

7y1 + 4y2 + 4y3 - 3eiωt
-6y1 - 4y2 - 7y3
-2y1 - y2 +2y3 + 3eiωt

4. Jul 4, 2013

5. Jul 4, 2013

### szynkasz

It can be solved by successive substitutions:

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

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.

6. Jul 4, 2013

### 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.

Last edited by a moderator: Jul 4, 2013