A diabetic patient receives insulin at constant rate from an implanted insulin pump. Insulin has first order elimination kinetics, so the amount of insulin in the blood will obey the given differential equation, where a> 0 is the rate at which insulin is released into their blood by the pump and k, is the fraction of insulin removed from the blood in one unit of time. dM d =a -k, M (a) Assuming M(0) = 0 (i.e., there is no insulin present in the patient's blood at time t=0), solve the differential equation to find M(t) as a function of t. (b) Find the limit of M(t) as t→0o. (c) Assume that a (the rate of release from the pump) is 1 IU/hr and M(t) → 0.2 IU as t+0o. Calculate k, , the rate of insulin elimination. (a) M(t) =

A diabetic patient receives insulin at constant rate from an implanted insulin pump. Insulin has first order elimination kinetics, so the amount of insulin in the blood will obey the given differential equation, where a> 0 is the rate at which insulin is released into their blood by the pump and k, is the fraction of insulin removed from the blood in one unit of time. dM d =a -k, M (a) Assuming M(0) = 0 (i.e., there is no insulin present in the patient's blood at time t=0), solve the differential equation to find M(t) as a function of t. (b) Find the limit of M(t) as t→0o. (c) Assume that a (the rate of release from the pump) is 1 IU/hr and M(t) → 0.2 IU as t+0o. Calculate k, , the rate of insulin elimination. (a) M(t) =

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

A diabetic patient receives insulin at constant rate from an implanted insulin pump. Insulin has first order elimination kinetics, so the amount of insulin in the blood will obey the given
differential equation, where a> 0 is the rate at which insulin is released into their blood by the pump and k, is the fraction of insulin removed from the blood in one unit of time.
dM
d =a -k, M
(a) Assuming M(0) = 0 (i.e., there is no insulin present in the patient's blood at time t=0), solve the differential equation to find M(t) as a function of t.
(b) Find the limit of M(t) as t→0o.
(c) Assume that a (the rate of release from the pump) is 1 IU/hr and M(t) → 0.2 IU as t+0o. Calculate k, , the rate of insulin elimination.
(a) M(t) =

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