Consider the (first order quasilinear) PDE $$ u \frac{\partial u}{\partial x} + (y+1) \frac{\partial u}{\partial y} = u \hspace{10mm} x \in \mathbb{R}, \hspace{4mm} y \in \left( 0, \hspace{2mm} \frac{1}{3} \right) $$ subject to $u(x,0) = -3x$ for $x \in \mathbb{R}$.
How might I find the solution to this PDE in parametric form? I have attempted to do this as follows:
The characteristic and compatibility equations associated with this PDE are $$ \frac{dx}{ds} = u \hspace{10mm} \frac{dy}{ds} = y+1 \hspace{10mm} \frac{du}{ds} = u $$ We may directly solve the second of these equations as $$ \frac{dy}{ds} = y+1 \hspace{5mm} \Rightarrow \hspace{5mm} y = y(s) = \frac{e^s}{c_1} - 1 $$ and for the third of these equations $$ \frac{du}{ds} = u \hspace{5mm} \Rightarrow \hspace{5mm} u = u(s) = \frac{e^s}{c_2} $$ and then, using the above, for the first of these equations we have $$ \frac{du}{ds} = u = \frac{e^s}{c_2} \hspace{5mm} \Rightarrow \hspace{5mm} x = x(s) = \frac{e^s}{c_2} + c_3 $$
However, I am unsure of where to go from here. Any advice for this would be appreciated.