Solving differential equation from variational principle

[JulieK]

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.

 

[Coelum]

JulieK,
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.