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 dt = 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→o. (c) Assume that a (the rate of release from the pump) is 5 IU/hr and M(t) → 0.2 IU as t→o0. Calculate k1, the rate of insulin elimination. (a) M(t) = U %3D (b) lim→M(t) = 00 (c) k, = | (Type an integer or decimal rounded to three decimal places as needed.) %3D

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ISBN:9780470458365
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Chapter2: Second-order Linear Odes
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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
- = a -k, M
dt
(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-00.
(c) Assume that a (the rate of release from the pump) is 5 IU/hr and M(t) → 0.2 IU as t+0o. Calculate k1, the rate of insulin elimination.
(a) M(t) =
(b) lim M(t) =
(c) k, =(Type an integer or decimal rounded to three decimal places as needed.)
Transcribed Image Text: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 - = a -k, M dt (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-00. (c) Assume that a (the rate of release from the pump) is 5 IU/hr and M(t) → 0.2 IU as t+0o. Calculate k1, the rate of insulin elimination. (a) M(t) = (b) lim M(t) = (c) k, =(Type an integer or decimal rounded to three decimal places as needed.)
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