I am currently working on solving the p.d.e.
$$ \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y} = 1 $$
with the initial condition
$$ u = 0 \text{ on } x + y = 0 $$
using the method of characteristics. Thus, so far, I have obtained a system of characteristic equations
$$ \begin{cases} \frac{dx}{dt} = 1 \implies x = t + x_0\\ \frac{dy}{dt} = 1 \implies y = t + y_0\\ \frac{du}{dt} = 1 \implies u = t + u_0 \end{cases} $$ However, I'm unsure how to proceed to encorporate the initial conditions or solve for $u(x, y)$ explicitly. Any help is appreciated.
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