Functional derivative of the functional $\int_{x_0}^{x_1} \left( y'^4(x) + e^{2y(x)} \right) \, \mathrm{d}x$

Question

How to find functional derivative of the following functional?

$$J(y) = \int_{x_0}^{x_1} \left( y'^4(x) + e^{2y(x)} \right) \, \mathrm{d}x$$

Answer 1

The functional derivative of a functional of the form

$$J(y) = \int_{x_0}^{x_1} L(x,y,y')\mathrm{d}x$$ is the LHR of the Euler-Lagrange equation: $$\frac{\delta J}{\delta y}=\frac{\partial L}{\partial y}-\frac{d}{dx}\frac{\partial L}{\partial y'}$$

so, in your case: $$\frac{\delta J}{\delta y}=2e^{2y}-4\frac{d}{dx}y'^3$$

Answer 2

Using SymPy to compute the Euler-Lagrange equation:

>>> x = Symbol('x')
>>> y = Function('y')(x)
>>> L = (Derivative(y,x))**4 + exp(2*y)
>>> euler_equations(L,y,x)
                          2   2           
    2*y(x)      /d       \   d            
[2*e       - 12*|--(y(x))| *---(y(x)) = 0]
                \dx      /    2           
                            dx            

Thus, the functional derivative is

$$2 e^{2 y} - 12 (y')^{2} y''$$