Solving the Euler-Lagrange Equation for Calculus of Variations

[insynC]

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

Homework Statement

Solve the Euler equation to make the following integral:

(integral from x1->x2)

∫ [(y')² + y²] dx

Homework Equations

Euler-Lagrange equation

∂F/∂y - d/dx (∂F/∂y') = 0

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

 

[Avodyne]

AFAIK this is the best you can do. You can't eliminate the dF/dy term in general.

 

[insynC]

Cheers. I'm a bit rusty on my calculus, but is there anything wrong with the step:

d/dx(2y') = 2y''

My lecturer made note to be careful of the d/dx part, but I don't think that applies here, just want to check.

 

[Dick]

That's just fine.