# Solving differential equation from variational principle

• Mar 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
$\displaystyle \frac{\partial}{\partial x}\left(\frac{1}{\sqrt{y}}\frac{dy}{dx}\right)=0$

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

where $\displaystyle f\left(y\right)$ is a function of $\displaystyle y$. I would like to get
$\displaystyle y$ as a function of $\displaystyle x$ but the problem is that I don't know the
form of $\displaystyle f\left(y\right)$.
My question, how to solve this equation to get $\displaystyle y$ as a function
of $\displaystyle x$. Is it possible to guess the form of $\displaystyle f\left(y\right)$, e.g.
from the bounday conditions.
• Mar 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.