2nd order system of linear DE through matrix methods

[iqjump123]

Homework Statement

obtain the solution of the coupled system of equations
d²X₁+2X₁=X₂
d²X₂+2X₂=X₁

Homework Equations

,

The Attempt at a Solution

I envisioned encountering this equation using matrix methods, as outlined in this website, since it was much easier than substitution, differentiation, etc:
http://tutorial.math.lamar.edu/Classes/DE/SystemsDE.aspx

However, unlike the method shown in the webpage above, my equation it is a 2nd order DE. Therefore, can I follow the steps exactly as outlined in this example involving only a first order system of DE?

If that is the case, then can I use the same method for the system of DE I wrote below?
https://www.physicsforums.com/showthread.php?t=517819
This problem is a little bit more different, because it contains a 2nd order AND a first order DE .

Thanks for the continued support!

 

[stratusfactio]

Let me clean that up for you:

$d^2x_1+2x_1=x_2$
$d^2x_2+2x_2=x_1$

 

[Ray Vickson]

Homework Statement

obtain the solution of the coupled system of equations
d²X₁+2X₁=X₂
d²X₂+2X₂=X₁

Homework Equations

,

The Attempt at a Solution

I envisioned encountering this equation using matrix methods, as outlined in this website, since it was much easier than substitution, differentiation, etc:
http://tutorial.math.lamar.edu/Classes/DE/SystemsDE.aspx

However, unlike the method shown in the webpage above, my equation it is a 2nd order DE. Therefore, can I follow the steps exactly as outlined in this example involving only a first order system of DE?

If that is the case, then can I use the same method for the system of DE I wrote below?
https://www.physicsforums.com/showthread.php?t=517819
This problem is a little bit more different, because it contains a 2nd order AND a first order DE .

Thanks for the continued support!

Set X3 = DX1 and X4 = DX2. Your system becomes:
DX1 = X3
DX2 = X4
DX3 = X2 - 2X1
DX4 = X1 - 2X2
so if V = column(X1 X2 X3 X4) you have DV = AV, where
A=[[0 0 1 0],[0 0 0 1],[-2 1 0 0],[1 -2 0 0]] = [row1,row2,row3,row4].

RGV

 

[iqjump123]

Set X3 = DX1 and X4 = DX2. Your system becomes:
DX1 = X3
DX2 = X4
DX3 = X2 - 2X1
DX4 = X1 - 2X2
so if V = column(X1 X2 X3 X4) you have DV = AV, where
A=[[0 0 1 0],[0 0 0 1],[-2 1 0 0],[1 -2 0 0]] = [row1,row2,row3,row4].

RGV

Hey RGV-

Thanks so much for your help!

By the way- matrix methods will be harder now I guess-
Trying to get eigenvalues and eigenvectors of a 4X4 matrix will be quite tough
(if I have just a calculator, pen, and paper).

So, am I correct to assume that standard substitution and DE solving methods are what has to be used to solve higher order linear system of ODE? (Solvable ones, that is)

thanks again.

iqjump123Update- Amongst different methods to solve this problem, I took the method of solving ODE through integrating factors, and got the answer that way. Thanks for the help!