Find a curve passing through $\left(0,0\right)$ and $\left(1,1\right)$ that is an extremal for the functional $\displaystyle{{\rm J}\left(x,y,y'\right) = \int\left\{\left[y'(x)\right]^{2} + 12x\,y\left(x\right)\right\}\,{\rm d}x}$
Hey, I'm not sure how to start this problem.. help is greatly appreciated! thanks!
$\newcommand{\+}{^{\dagger}}% \newcommand{\angles}[1]{\left\langle #1 \right\rangle}% \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}% \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}% \newcommand{\dd}{{\rm d}}% \newcommand{\isdiv}{\,\left.\right\vert\,}% \newcommand{\ds}[1]{\displaystyle{#1}}% \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}% \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}% \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}% \newcommand{\ic}{{\rm i}}% \newcommand{\imp}{\Longrightarrow}% \newcommand{\ket}[1]{\left\vert #1\right\rangle}% \newcommand{\pars}[1]{\left( #1 \right)}% \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}}% \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}% \newcommand{\sech}{\,{\rm sech}}% \newcommand{\sgn}{\,{\rm sgn}}% \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}}% \newcommand{\verts}[1]{\left\vert #1 \right\vert}% \newcommand{\yy}{\Longleftrightarrow}$ $$ 0 = \delta{\rm J} = \int\braces{2y'\delta y' + 12 x\delta y}\,\dd x = \int\braces{\totald{\pars{2y'\delta y}}{x} - \delta y\bracks{2y'' - 12 x}}\,\dd x $$ $$ 2y'' - 12 x = 0\quad\imp\quad y\pars{x} = x^{3} + Ax + B $$ With $y\pars{0} = 0$ and $y\pars{1} = 1$ we get $A = B = 0$. Then $$ \color{#0000ff}{\large y\pars{x} = x^{3}} $$