Solve by Method of Characteristics

1. Oct 5, 2011

syj

1. The problem statement, all variables and given/known data
$u\frac{\delta u}{\delta x}$+$\frac{\delta u}{\delta y}$ =1

$u|_{x=y}=\frac{x}{2}$

2. Relevant equations

the characteristic equations to solve are:

$\frac{dx}{ds}=u$

$\frac{dy}{ds}=1$

$\frac{du}{ds}=1$

3. The attempt at a solution

I got the following equations from the characteristic equations:

$\frac{dx}{ds}=u$ means $\frac{d^2x}{ds^2}=\frac{du}{ds}=1$
after integrating twice i get $x=\frac{1}{2}s^2+ks+x_0$

$y = s +y_{0}$

$u = s +u_{0}$

and here is where I am now stuck.
I also dont know how to apply the condition given : $u|_{x=y}=\frac{x}{2}$

Last edited: Oct 5, 2011