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2nd order nonlinear ODE help needed

  1. Dec 13, 2011 #1
    Hel(lo, p)

    I hope you're doing fine

    I'm stuck with the following:

    [itex] y'' = -1/(y^2)[/itex]

    I tried guessing functions (exponentials, roots, trigs... ) , but none worked, I haven't had any DE course, so I don't have specific steps to employ,

    I appreciate your help,
    Thanks in advance
     
  2. jcsd
  3. Dec 13, 2011 #2
    the case where [itex]y^{''}=f(y,y^{'})[/itex] is called an autonomous equation (x does not occur directly in the right-hand side).

    It can be solved by performing a simple transformation (this transformation follows from a translational symmetry of the ODE)

    Let [itex]y^{'}=z[/itex], then
    [itex]y^{''}=z^{'}=\frac{dz}{dx}=\frac{dz}{dy}\frac{dy}{dx}=\frac{dz}{dy}z[/itex]
    and the equation can be written as:
    [itex]z\frac{dz}{dy}=-\frac{1}{y^2}[/itex]
    [itex]\int z dz=-\int \frac{1}{y^2}dy[/itex],
    then transform back to the original variable y and integrate again.
     
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