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A Property of an Autonomous ODE

  1. Jan 24, 2009 #1
    Hi!

    I wonder how to prove that if y(t)=sin(t) solves an autonomous ODE f(y,y',...,y^(n))=0, then x(t)=cos(t) is also a solution.

    I mean I'm a bit distracted by the fact that all derivatives of y are present here. For example in the equation for a pendulum there are just y and y'' and a special functional dependence between them and that's why it works, isn't it?

    In the above case I know almost nothing about the function f. So how to proceed?
     
  2. jcsd
  3. Jan 24, 2009 #2

    CompuChip

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    Maybe you can use that cos(x) = sin(x + pi / 2)?
     
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