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I Convert complex ODE to matrix form

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  1. Nov 25, 2017 #1

    SeM

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    Hi, I have the following complex ODE:

    aY'' + ibY' = 0

    and thought that it could be written as:

    [a, ib; -1, 1]

    Then the determinant of this matrix would give the form

    a + ib = 0

    Is this correct and logically sound?

    Thanks!
     
  2. jcsd
  3. Nov 25, 2017 #2

    haruspex

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    As far as I am aware, writing in matrix form is for when you have a system of differential equations in correspondingly multiple variables. I do not see how it can be applied to a single equation. On what would you base the choice of coefficients for the second row?
     
  4. Nov 27, 2017 #3
    Hey, SeM,

    I don't understand how you arrive at the second row in the matrix. Maybe, what you want to convert the second order differential equation into a system of two differential equations. This can be done by introducing
    $$W(x)=Y'(x),$$
    leading to the equations
    $$aW'(x)+ibW(x)=0, \quad Y'(x)=W(x)$$

    Indeed, both of your equations, i.e.
    $$aY''(x)+ibY'(x)=0, Y''(x)-Y'(x)=0$$
    if
    $$a+ib=0,$$ but this is only a special case and it doesn't follow from your stated differential equation.
     
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