Find the function $y$ with which the functional $$J=\int_{x_0}^{x_1}(2xy+(y^{\prime\prime\prime})^2)dx$$ has extremum.
Here's what I've been suggested so far:
we use $\delta J=0$ with any $\delta y$, then we need to write down the Euler equation $$\frac {dF}{dy}-\frac{d}{dx}\left(\frac{\partial F}{\partial y^{\prime}}\right)+\frac{d^2}{dx^2}\left(\frac{\partial F}{\partial y^{\prime\prime}}\right)-\frac{d^3}{dx^3}\left(\frac{\partial F}{\partial y^{\prime\prime\prime}}\right)=0.$$ $F$ is the integerand here. Some of the terms are $0$, so they simplify. We need to solve the differential equation. The answer should be in the form of $y=$, with constants. This is all I know so far, however I am struggling to connect the dots. I'd very much appreciate if someone could help me see the full solution.