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Math Help - 2nd order homogenous in to 1st order linear systems

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by DistantCube View Post
    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
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