\documentclass[a4paper,12pt]{article}
\newcommand{\ds}{\displaystyle}
\newcommand{\un}{\underline}
\newcommand{\undb}{\underbrace}
\newcommand{\pl}{\partial}
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\begin{document}

{\bf Question}

If $yu_x+u_y=0$ for all $x,\ y$ and $u(x,0)=exp(-x)$ for all $x$,
find $u(x,y)$.


\medskip

{\bf Answer}

From Lagrange's equations,

$\ \ \ds\frac{dx}{d\xi}=y,\ \ds\frac{dy}{d\xi}=1,\
\ds\frac{du}{d\xi}=0$

$\Rightarrow \ds\frac{dx}{dy}=\ds\frac{y}{1},\
\ds\frac{du}{d\xi}=0$

$\Rightarrow x=\ds\frac{y^2}{2}+b,\ u=a$ ($b,\ a$ consts)

$\Rightarrow$

$(\star) \left\{\begin{array} {rcl} x-\ds\frac{y^2}{2} & = &
const\\ u & = & cpmst \end{array} \right.$

$\Rightarrow \un{u=f\left(x-\ds\frac{y^2}{2}\right)}$

If $u=e^{-x}$ at $y=0$, then

$e^{-x}=f(x)$ i.e., $f(\eta)=e^{-\eta}$ for any $\eta$.

$\Rightarrow \un{u=e^{-(x^2-\frac{y^2}{2})}}$


\end{document}
