Characteristic curves for constant velocity field

[rexasaurus]

1. Show that, if the velocity field (V) is a fixed (spatially constant) vector, then the characteristic curves will be a family of parallel-straight lines.



2. u_t+V₁u_x+V₂u_y=f
f=S-[dell dotted with V]u

characteristic curves:
dX/dt=V₁(X,Y) & dY/dt=V₂(X,Y)


3. really looking for help on how to get started on this. any suggestions would be appreciated.

 

[Coto]

Since V is fixed (i.e. $V_1(x,y) = C_1; V_2(x,y) = C_2$, you can solve the parametric equations for the characteristic curves explicitly, and obtain X(t), and Y(t). What types of curves are these?