3rd order homogeneous Linear ODE matrix transformation-how to write the answer

[pat666]

3rd order homogeneous Linear ODE matrix transformation---how to write the answer

Homework Statement

Transform this 3rd order homogeneous linear ODE with constant coefficients using matrix notation.
$$y'''+7y''+6y'+3y=0$$

Homework Equations

The Attempt at a Solution

My answer is
$$x'=AX$$
$$x=[x_1,x_2,x_3]^T$$
$$A=Mat[[1,0,0][0,1,0][-7,-6,-3]]$$

x'=
(x_1)
(x_2)
(-7x_1-6x_2-3x_3)

My question is do I leave in this form and also is it correct?

Thanks

 

[HallsofIvy]

i would strongly advise you to write out precisely what substitutions you are making. It will make it much easier for all, including yourself, to see what you are doing and whether or not it is correct (not to mention shock and amaze your teacher!).

You appear to be saying that
$$x= \begin{bmatrix}x_1 \\ x_2\\ x_3\end{bmatrix}$$
where
$x_1= y$, $x_2= y'$, and $x_3= y''$.

Then the given equation becomes $x_3'+ 7x_3+ 6x_2+ 3x_1= 0$ which is the same as $x_3'= -3x_1- 6x_2- 7x_3$.
Of course, you also have $x_1'= x_2$ and $x_2'= x_3$ so your equation, in matrix form, is
$$x'= \begin{bmatrix}x_1' \\ x_2' \\ x_3'\end{bmatrix}$$$$= \begin{bmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ -3 & -6 & -7\end{bmatrix}\begin{bmatrix}x_1 \\ x_2 \\ x_3 \end{bmatrix}$$

which is, I believe, what you have.

 

[pat666]

Yes that's exactly what I meant. Sorry for the illegible post, I didn't know the LaTex for Matrices. How would you advice me to write the answer? I have posted what your final matrix equation comes out to: 3rd row: $$-7x_1-6x_2-3x_3$$
which I think would go back to $$y=-7y-6y'-3y''$$?Thanks

 

[pat666]

Hi again HallsofIvy, one simple question, Is my solution to this ODE $x_3'= -3x_1- 6x_2- 7x_3$. if so why did I need matrices and what use is that?? I also forgot the final bit of the question.
Question:
Transform this 3rd order homogeneous linear ODE with constant coefficients using matrix notation.
ODE
into a system of first oder ODE's.

Have I done this?

Thanks

 

[HallsofIvy]

Hi again HallsofIvy, one simple question, Is my solution to this ODE $x_3'= -3x_1- 6x_2- 7x_3$. if so why did I need matrices and what use is that?? I also forgot the final bit of the question.
Question:
Transform this 3rd order homogeneous linear ODE with constant coefficients using matrix notation.
ODE
into a system of first oder ODE's.

Have I done this?

Thanks

First, that's not a solution to the differential equation though it is a (partial) solution to the question asked. However, just saying "$x_3'= -3x_1- 6x_2- 7x_3$" is not the entire solution because it does not give the important equations for $x_1$ and $x_2$.

You "need" the matrix equations because that was what the problem told you to do!

Yes, converting to a system of first order equations would be: $x_1'= x_2$, $x_2'= x_3$, and $x_3'= -3x_1- 6x_2-7x_3$.

 

[pat666]

So I was not actually looking for a "solution" to this ODE?

Thanks HoI