Just did this in class today and was doing a problem to see if I understood it and I'm not sure I did. Thanks for any help(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Solve the Euler equation to make the following integral:

(integral from x1->x2)

∫ [(y')² + y²] dx

2. Relevant equations

Euler-Lagrange equation

∂F/∂y - d/dx (∂F/∂y') = 0

3. The attempt at a solution

Clearly F = (y')² + y²

In class we had been rearranging the integral so that ∂F/∂y = 0, which made the problem much simpler. I was unable to do this here as there was no apparent way to introduce ds to swap y' for x'. This was my concern as, although I didn't think all equations would be this simple, I thought most would reduce.

So instead I proceeded as follows:

∂F/∂y - d/dx (∂F/∂y') = 2y - d/dx (2y') = 2y - 2y'' = 0 => y'' = y

And this is easy enough to solve. But I'm concerned I've made a mistake getting there.

Have I made a mistake up to here or is there a better way to proceed?

Thanks

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# Homework Help: Calculus of Variations

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