I just want to verify and check my understanding to see if I can do this right.
a)$ u_x + u_y = 0$
This is linear since: $(u+v)_x + (u+v)_y = u_x +u_y + v_x+v_y $
and $c(u_x+u_y) = cu_x + cu_y$
b) $u_x + yu_y = 0 $
This is linear since:
$(u+v)_x + y(u+v)_y = u_x+v_x + yu_y + yv_y $
and $c(u_x+yu_y) = cu_x+ ycu_y$
c) $u_x + uu_y = 0$
Linear again, I don't see the fail of linearity on this example as well.