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Solving differential equation from variational principle

  1. Mar 26, 2013 #1
    I have the following differential equation which I obtained from Euler-Lagrange
    variational principle
    [itex]\frac{\partial}{\partial x}\left(\frac{1}{\sqrt{y}}\frac{dy}{dx}\right)=0[/itex]

    I also have two boundary conditions: [itex]y\left(0\right)=y_{1}[/itex] and
    [itex]y\left(D\right)=y_{2}[/itex] where [itex]D[/itex], [itex]y_{1}[/itex] and [itex]y_{2}[/itex] are known
    I assume that I should integrate once to get

    where [itex]f\left(y\right)[/itex] is a function of [itex]y[/itex]. I would like to get
    [itex]y[/itex] as a function of [itex]x[/itex] but the problem is that I don't know the
    form of [itex]f\left(y\right)[/itex].
    My question, how to solve this equation to get [itex]y[/itex] as a function
    of [itex]x[/itex]. Is it possible to guess the form of [itex]f\left(y\right)[/itex], e.g.
    from the bounday conditions.
  2. jcsd
  3. Mar 27, 2013 #2
    I am not sure I understand: why are you applying a partial differential operator to what looks like a function of one variable? If that is just a typo, then your equation gets much easier: you just equate the term inside the parenthesis to a constant and get an ODE that is integrable by part.
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