= Solution
{c}
The <method of characteristics> replaces the <first-order partial differential equation> by the <ordinary differential equations>
$$
\dot x=x,\qquad \dot y=2y+\beta x,\qquad \dot u=x+2.
$$
An <integrating factor> for the second equation gives $d(e^{-2s}y)/ds=\beta C e^{-s}$, so their general <solution of a differential equation> is
$$
\boxed{x=C e^s,\qquad y=D e^{2s}-\beta C e^s,\qquad u=C e^s+2s+E.}
$$
Here $C,D,E$ are real constants. For $C\ne0$, eliminating $s$ gives the <characteristic projections>
$$
\boxed{y=Kx^2-\beta x,\qquad K=D/C^2,}
$$
each restricted to either $x>0$ or $x<0$. These are arcs of <parabolas>, with the straight-line exception $K=0$. Along an oriented <characteristic curve>, $|x|$ increases with $s$.
For $C=0$, the <characteristic projections> lie on $x=0$: $y=D e^{2s}$ describes either vertical half-line when $D\ne0$. For $C=D=0$ the projection is the single point $(0,0)$, although $u=2s+E$ still changes. This last lifted <characteristic curve> cannot be contained in the <function graph> of a <classical solution>: at $(0,0)$ the original <partial differential equation> would require $0=2$.
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