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Math Help - Solving differential equation from variational principle

  1. #1
    Junior Member
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    Solving differential equation from variational principle

    I have the following differential equation which I obtained from Euler-Lagrange
    variational principle
    \frac{\partial}{\partial x}\left(\frac{1}{\sqrt{y}}\frac{dy}{dx}\right)=0

    I also have two boundary conditions: y\left(0\right)=y_{1} and
    y\left(D\right)=y_{2} where D, y_{1} and y_{2} are known
    numbers.
    I assume that I should integrate once to get
    \frac{1}{\sqrt{y}}\frac{dy}{dx}=f\left(y\right)

    where f\left(y\right) is a function of y. I would like to get
    y as a function of x but the problem is that I don't know the
    form of f\left(y\right).
    My question, how to solve this equation to get y as a function
    of x. Is it possible to guess the form of f\left(y\right), e.g.
    from the bounday conditions.
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  2. #2
    Senior Member
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    Re: Solving differential equation from variational principle

    Since y is a function of x, then f(y) is function of x. It's derivative is 0, hence f(y)=constant.
    Then, solve dy/dx=c*sqrt(y), which is a separable ODE.
    Thanks from JulieK
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