Thread: 2nd order homogenous in to 1st order linear systems

I need help to understand this. The book (Kreyzig) is very vague to me in this area. I have a few questions to do, but all I ask is you help me with this general case so I can understand how to do the rest of my questions, and hopefully I will be able to do them on my own. [as an aside, the book shows the case of y'=Ay, the case I'm dealing with is of 2nd order...i'm lost even though its basic]

Convert the following second-order homogeneous linear DE into ho-
mogeneous linear systems of first-order ODEs, for which determine the real general solution:

y''- (k^2)y = 0, where k is not 0.

Originally Posted by DistantCube

I need help to understand this. The book (Kreyzig) is very vague to me in this area. I have a few questions to do, but all I ask is you help me with this general case so I can understand how to do the rest of my questions, and hopefully I will be able to do them on my own. [as an aside, the book shows the case of y'=Ay, the case I'm dealing with is of 2nd order...i'm lost even though its basic]

Convert the following second-order homogeneous linear DE into ho-
mogeneous linear systems of first-order ODEs, for which determine the real general solution:

y''- (k^2)y = 0, where k is not 0.

Introduce the state vector Y=(y, dy/dt)' (where the ' denotes the
transpose, as I normally work with column vectors).

Then:

dY/dt = (dy/dt, d^2y/dt^2)' = (dy/dt, k^2 y)'

....... = AY,

where :

Code:

A = [0   ,1]
    [k^2, 0]

RonL