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Change of dependent variable in a DE

  1. Jun 23, 2009 #1
    Hi all, I have what should hopefully be a quick question. Given an ODE of the following form (sorry no tex)
    y = y(x) with y' defined as differentiation wrt (with respect to) x

    y'' + y' + y = 0

    and I want to make a change a variables A = x/a (for some constant a) so that we define a new dependent variable as Y(A) = y(ax).

    I now want to rewrite the ODE with respect to Y and A.
    Y = Y(A) with Y* defined as differentiation wrt A

    I am not sure how this works out, but I know that the answer should look like

    1/a^2 Y** + 1/a Y* + Y = 0

    I know this isnt a hard question but I'm just not seeing it. Thanks in advance for help.


    I found a way for this to work out, although it seems a bit convoluted. Input still welcome though.
    Last edited: Jun 23, 2009
  2. jcsd
  3. Jun 24, 2009 #2


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    Use the chain rule. If A= x/a, then dA/dx= 1/a (kind of wish you had chosen some other letters!:smile:).

    [tex]\frac{dY}{dx}= \frac{dY}{dA}\frac{dA}{dx}= \frac{1}{a}\frac{dY}{dx}[/tex]

    [tex]\frac{d^2Y}{dx^2}= \frac{d}{dx}\left(\frac{dA}{dx}\right)= \frac{d}{dx}\left(\frac{1}{a}\frac{dY}{dA}\right)[/tex
    [tex]= \frac{1}{a}\frac{d}{dx}\left(\frac{dY}{dA}\right)= \frac{1}{a}\left(\frac{1}{a^2}\frac{d^2Y}{dA^2}[/tex]
    [tex]= \frac{1}{a^2}\frac{d^2Y}{dA^2}[/tex]

    So your equation is
    [tex]\frac{1}{a^2}\frac{d^2Y}{dA^2}+ \frac{1}{a}\frac{dY}{dA}+ Y=0[/tex]
    If you like you can multiply on both sides by [itex]a^2[/itex] and get
    [tex]\frac{d^2Y}{dA^2}+ a\frac{dY}{dA}+ a^2Y= 0[/tex].

    Notice that this would work for A being a function of x also, though then the derivatives become more complicated.
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