Converting a Single ODE to Matrix Form for Eigenvalue Analysis

[SeM]

Hi, I have the following ODE:

aY'' + bY' + c = 0

I would like to convert it to a matrix, so to evaluate its eigenvalues and eigenvectors. I have done so for phase.plane system before, however there were two ODEs there. In this case, there is only one, so how does this look like in a matrix form given that this is non homogenous? Is it as such:

\binom{a \ \ 1}{-b \ \ 1}\binom{c}{0}}

Or how is it correct to convert ONE ODE to matrix form, and from there study it?

 

[bigfooted]

you can write a second order ode as a system of two first order ode's by introducing new variables:
$y_1 = Y$
$y_2 = \frac{dY}{dx}$

such that the original ode becomes:
$a\frac{dy_2}{dx} + by_1 +c = 0$
with the auxiliary ode:
$y_2 = \frac{dy_1}{dx}$

 

[SeM]

Thanks!

 

[SeM]

Is there any chance of getting the constant c in the matrix form at all, or should a matrix form rather give the homogenous form in order to solve the eigenvalues?

 

[bigfooted]

Well, do you think the eigenvalues will change by the constant c?

 

[SeM]

No, they won't indeed! Thanks!