A mathematical model for the rate at which a drug disseminates into the bloodstream is given by

dx/dt = r - kx,

where r and k are positive constants. The function x(t) describes the concentration of the drug in the bloodstream at time t.

(a) Since the DE is autonomous, use the phase portrait concept of Section 2.1 to ­find the limiting value of x(t) as t → ∞.

(b) Solve the DE subject to x(0) = 0. Sketch the graph of x(t) and verify your prediction in part (a). At what time is the concentration one-half this limiting value?