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Substitution question

  1. Apr 17, 2010 #1
    1. The problem statement, all variables and given/known data
    I'm reading a book where they do the following steps which I don't understand:
    We have a DE:
    b^2 * y'' = axy
    put t = b^(-2/3) a ^(1/3) x
    then somehow get (d^2 y)/dt^2 = ty

    2. Relevant equations

    3. The attempt at a solution
    I tried messing with chain rule but got nowhere.
  2. jcsd
  3. Apr 17, 2010 #2


    User Avatar
    Science Advisor

    Yes, the "chain rule" is the way to go.

    If [itex]t= b^{-2/3}a^{1/3}x[/itex] then [itex]dt/dx= b^{-2/3}a^{1/3}[/itex] and [itex]x= b^{2/3}a^{-1/3}t[/itex]

    [tex]\frac{dy}{dx}= \frac{dy}{dt}\frac{dt}{dx}= b^{-2/3}a^{1/3}\frac{dy}{dt}[/tex]

    Doing that again,
    [tex]\frac{d^2y}{dx^2}= b^{-4/3}a^{2/3}\frac{d^2y}{dt^2}[/tex]

    Now, we have
    [tex]b^{2- 4/3}a^{2/3}\frac{d^2y}{dt^2}= a^{1- 1/3}b^{-2/3}ty[/tex]
    [tex]b^{-2/3}a^{2/3}\frac{d^2y}{dt^2}= a^{2/3}b^{-2/3}ty[/tex]

    [tex]\frac{d^2y}{dt^2}= ty[/tex]

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