A hospital patient is receiving pain medication through an IV. The drug enters t

Question

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.

Explanation / Answer

The rate at which the drug leaves the bloodstream:=ky where k is a constant

dy/dt=r-ky

dy/dt+ky-r=0 homogenous

dy/(r-ky)=dt

ln(r-ky)=-kt+c

r-ky=Ce^-kt

ky=r-Ce^-kt

y=r/k - (C/k).e^-kt

putting y(0)=0

0=r/k-C/k

C=r

y=(r/k)(1-e^-kt)

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