stationary function of an integral

Question

Find the stationary function $y=y(x)$ of the integral $\int_o^4[xy'-(y')^2]dx$ satisfying the conditions $y(0)=0$ and $y(4)=3$.

I don't know what a stationary function is.

Can you anyone suggest me where i can find some theory related to this??

Answer 1

$\newcommand{\+}{^{\dagger}}% \newcommand{\angles}[1]{\left\langle #1 \right\rangle}% \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}% \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}% \newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}% \newcommand{\dd}{{\rm d}}% \newcommand{\ds}[1]{\displaystyle{#1}}% \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}% \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}% \newcommand{\fermi}{\,{\rm f}}% \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}% \newcommand{\half}{{1 \over 2}}% \newcommand{\ic}{{\rm i}}% \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow}% \newcommand{\isdiv}{\,\left.\right\vert\,}% \newcommand{\ket}[1]{\left\vert #1\right\rangle}% \newcommand{\ol}[1]{\overline{#1}}% \newcommand{\pars}[1]{\left( #1 \right)}% \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}}% \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}% \newcommand{\sech}{\,{\rm sech}}% \newcommand{\sgn}{\,{\rm sgn}}% \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}}% \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ $\ds{L \equiv xy' - y'^{2}\,,\qquad\totald{}{x}\pars{\totald{L}{y'}} = \partiald{L}{y}}$ $$ \totald{}{x}\pars{x - 2y'} = 0\,,\quad x - 2y' = -2C\,,\quad y' = \half\,x + C\,,\quad y = {1 \over 4}\,x^{2} + Cx + D $$ $$ {1 \over 4}\,0^{2} + C\times 0 + D = 0\,,\quad D = 0\,;\qquad\qquad {1 \over 4}\,4^{2} + C\times 4 = 3\,,\quad C = -\,{1 \over 4} $$ $$ \color{#0000ff}{\large y = {1 \over 4}\,x\pars{x - 1}} $$