Dealing with total derivative in Euler Equation

Question

I am trying to solve this:

$I = \int_0^1(x^2-y^2-(y')^2)$ using the euler equation: $\frac{d}{dx}[\frac{\partial F}{\partial y'}]-\frac{\partial F}{\partial y} =0$

and find the function y(x). So, I have:

$\frac{d}{dx}[2y']-2y=0$. How do I deal with the $\frac{d}{dx}$? Re-writing it in some form of partial derivatives?

Answer 1

Here is how

$$ \frac{d}{dx}[2y']-2y=0 \implies 2y''-2y=0\implies y''(x)-y(x)=0. $$

Now what's left is to solve the last ode.