A hospital patient is receiving pain medication through an IV. The drug enters the patient's bloodstream at a constant rate r. The rate at which the drug leaves the bloodstream is proportinal to the amount of drug present. Let t represent time (measured in minutes) and let y(t) represent the amount of the drug in the bloodstream at time t.
(a) Determine the first-order ODE that models this situation.
(b) Rewrite the ODE to show tht it is linear, and state whether it is homogeneous or non-homogeneous.
(c) Describe what form the solutions to this ODE will have.
(d) Solve to find a one-parameter family of solutions.
(e) Find a particular solution satisfying the initial condition y(0)=0.