Problem
Let
$$\frac{\partial z}{\partial x}+2\frac{\partial z}{\partial y}+3z=0$$
Find the characteristics associated with this PDE and find an explicit solution $z(x,y)$ which satisfies the initial condition $z=e^{x}$ on the line segment $\{(x,y):0 \leq x \leq 1, y=0 \}$.
State the region on the $(x,y)$-plane in which $z(x,y)$ is uniquely determined by the initial condition.
Progress
To find the characteristics:
$$\frac{dx}{ds}=1, \quad \frac{dy}{ds}= 2, \quad \frac{dz}{ds}=-3z$$ which yields that $$x=s+A, \quad y=2s+B, \quad z=Ce^{-3s}$$
Are we now OK to parametrise the line segment as $L=(t,0,e^t)$ using our initial condition, giving rise to explicit parametric solutions
$$\begin{cases} x=s+t \\ y=2s \\ z=e^{t-3s} \end{cases}$$ for $0 \leq t \leq 1$.
I'm not convinced that having $z=e^x$ allows us to parametrise the line segment in the way. Also, to establish where the solution is uniquely determined:
At $t=0$,
$$x=s, \quad y=2s, \quad \text{so } y=2x$$ and at $t=1$,
$$ x=s+1, \quad, y=2s, \quad \text{so } y=2x-2$$
Not sure how much guidance this offers.
Any help with the various parts of this would be much appreciated. Not sure what's right/wrong really. Regards as always, MM.