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Autonomous Second Order ODE

  1. Jul 14, 2009 #1
    Hi All,

    I was looking for the general solution of an equation as y(x)'' = f (y), and found the attached document on the web.

    It presents the solution in a way which I am not sure I understand. I tried to look at the trivila example y'' = - y, solution y = sin (x), but I am struggling in obtaining this result with the provided solution.

    If anybody could help, I would be the most obliged.

    All the Best

    Muzialis
     

    Attached Files:

  2. jcsd
  3. Aug 8, 2009 #2
    Hi,

    The trick is to define a new variable, y'=dy/dx=p(y(x)).
    Now y''=dp/dy*y'=dp/dy*p=f(y).
    Integrating now yields p^2=c+int(f(y))dy. The next step is to integrate dx=dy/p(y).

    Best, Mathador
     
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