I have the following differential equation which I obtained from Euler-Lagrange(adsbygoogle = window.adsbygoogle || []).push({});

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

numbers.

I assume that I should integrate once to get

[itex]\frac{1}{\sqrt{y}}\frac{dy}{dx}=f\left(y\right)[/itex]

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.

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# Solving differential equation from variational principle

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