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Change of variable for leibniz notation

  1. Jul 17, 2011 #1
    Hi

    I have a question for change of notation.

    Quote from text book:

    As an example of a singular problem on a finite interval, consider the equation
    xy'' + y' + λxy = 0, (6)
    or
    −(xy')' = λxy, (7)
    on the interval 0 < x < 1, and suppose that λ > 0. This equation arises in the study of
    free vibrations of a circular elastic membrane, and is discussed further in Section 11.5.
    If we introduce the new independent variable t defined by t =sqrt(λ)x, then
    dy/dx=sqrt(λ)*dy/dt,
    d2y/dx2 = λ*d2y/dt2.
    ...

    my question is the steps involved in deriving d2y/dx2 = λ d2y/dt2.

    My understanding in deriving dy/dx=sqrt(λ)*dy/dt is as follows:

    since dy/dt = dy/dx*dx/dt
    and dx/dt ->(using differentials) dt = sqrt(λ)dx -> dx/dt = 1/sqrt(λ) -> dy/dx = sqrt(λ)*dy/dt.

    However, how do you derive d2y/dx2 = λ d2y/dt2?
    I get 1=1 which is no help at all?

    Sorry if this is noobie, I'm kind of new.

    Also, is there a online notepad for practicing mathjax synatax? I like pretty equations.

    Thank you.
     
  2. jcsd
  3. Jul 18, 2011 #2

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    welcome to pf!

    hi progenitor! welcome to pf! :smile:

    (have a square-root: √ and try using the X2 icon just above the Reply box :wink:)
    use the chain rule, d/dx = d/dt dt/dx = d/dt 1/√λ :wink:

    (and you can practise your mathjax at https://www.physicsforums.com/mathjax/test/preview.html" [Broken] :smile:)
     
    Last edited by a moderator: May 5, 2017
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