$\displaystyle u_x + u_y + u = 1 $

$\displaystyle u = sin(x)$ on $\displaystyle y = x + x^2, x>0$

Let $\displaystyle x = s $ be the parameter along the initial line $\displaystyle y = x + x^2$

Solution:

Characteristic Equation:

$\displaystyle \frac{dx}{dt} = 1, x = t + s$

$\displaystyle \frac{dy}{dt} = 1, y = t + s +s^2 $

$\displaystyle \frac{du}{dt} = 1 - u, u(0) = sin(s) $

Hence:

$\displaystyle y = s + x^2$

$\displaystyle u = 1 + (sin(s) - 1)e^{-t}$

and the solution continues on...

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What I don't understand is how to get $\displaystyle u = 1 + (sin(s) - 1)e^{-t}$.

I sort of understand the $\displaystyle \frac{dx}{dt} = 1, x = t + s$ part, but can someone give me an explanation as well?

Thanks =]