# Solving differential equation from variational principle

• March 26th 2013, 09:29 AM
JulieK
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.
• March 26th 2013, 09:59 AM
JJacquelin
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.