How to set the variation of an integral to 0?

[phoneketchup]

So I have an integral:

$\delta W=\int_{-\Delta}^\Delta\left[x^2\left(\frac{d\xi}{dx}\right)^2−D_S\xi^2\right]dx$
Here $\xi$ is a function of $x$ and $D_S$ is a constant. $\Delta$ is just some small $x$. Now I need to set the variation of $\delta W$ to 0. Do do this I differentiated whatever is inside the bracket and set it to 0. I get:

$x^2\xi″+x\xi′−D_S\xi = 0$

However, the answer is:

$\frac{d}{dx}\left(x^2\frac{d\xi}{dx}\right)+D_S\xi = x^2\xi″+2x\xi′+D_S\xi = 0$

Where the primes are derivatives with respect to x. As you can see the difference is a factor of 2 in the middle term and that minus sign.

If anyone could point out where I am going wrong, it would be really appreciated.

Thanks!

 

[Orodruin]

You cannot simply differentiate the integrand with respect to the integration variable. You need to check out a textbook or lecture notes on variational calculus and apply Euler-Lagrange's equations.

 

[phoneketchup]

Thanks a lot! Got the answer!