How to find functional derivative of the following functional?

Question

How to find functional derivative of the following functional?

$$J(y) = \int_{x_0}^{x_1} \left( y^6(x) + 3 y^2(x) \right) \, \mathrm{d}x$$

Answer 1

You can find by the help of Euler-Lagrange Equations. partial derivative of $L$ w.r.t $y$ is $6y^5+6y$ and partial derivative of $L$ w.r.t $y'$ is zereo. So, Euler-Lagrange equation is $6y^5+6y$.

Answer 2

The Lagrangian is

$$\mathcal L (y) := y^6 + 3 y^2$$

and the functional derivative is

$$\mathcal L' (y) = 6 y^5 + 6 y = 6 y \, (y^4 + 1)$$