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\newcommand{\ds}{\displaystyle}
\newcommand{\un}{\underline}
\newcommand{\undb}{\underbrace}
\newcommand{\pl}{\partial}
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\begin{document}

{\bf Question}

Find the partial differential equation satisfied by $u(x,y)$ if

$$I=\ds\int \!\!\! \ds\int_S \{au^2_x+2bu_xu_y+cu^2_y+eu^2\}dxdy$$

where $a,\ b,\ c,\ e$ are functions of $x$, is to be stationary
subject to $u(x,y)$ taking specified values on the boundary of
$S$.


\medskip

{\bf Answer}

With $F=au_x^2+2bu_xu_y+cu_y^2+eu^2$ the E-L equation is

$\ds\frac{\pl F}{\pl u}-\ds\frac{\pl}{\pl x}\left(\ds\frac{\pl
F}{\pl u_x}\right)-\ds\frac{\pl}{\pl y}\left(\ds\frac{\pl F}{\pl
u_y}\right)=0$ with $a=a(x),\ b=b(x)$ etc.

$\Rightarrow \ds\frac{\pl}{\pl x}(2au_x+2bu_y)+\ds\frac{\pl}{\pl
y}(2bu_x+2cu_y)-2eu=0$

$\Rightarrow
au_{xx}+2bu_{xy}+cu_{yy}+\ds\frac{da}{dx}u_x+\ds\frac{db}{dx}u_y-eu=0$

\end{document}