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(a) Define the function C(t) = M(t)e – é, teR Show that C(t) = M'(t)e' + M(t)e – é (b) Use (a) and (5.5) to show that C'(t) = 0 for all t e R. (c) Use Corollary 2 to show that C(t) is a constant and, hence, M(1) = C(0) = –1. (d) Show that (c) implies that M(t)e – d = -1, and therefore: M(t) = 1 – e~'.

(a) Define the function C(t) = M(t)e – é, teR Show that C(t) = M'(t)e' + M(t)e – é (b) Use (a) and (5.5) to show that C'(t) = 0 for all t e R. (c) Use Corollary 2 to show that C(t) is a constant and, hence, M(1) = C(0) = –1. (d) Show that (c) implies that M(t)e – d = -1, and therefore: M(t) = 1 – e~'.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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(a) Define the function C(1) = M(t)d' – d', teR Show that C'(1) = M'(1)e + M(1)e – é (b) Use (a) and (5.5) to show that C'(t) = 0 for all t e R. (c) Use Corollary 2 to show that C(t) is a constant and, hence, M(t) = C(0) = –1. (d) Show that (c) implies that M(t)e – é = -1, and therefore: M(t) = 1 – e-'.
Transcribed Image Text:(a) Define the function C(1) = M(t)d' – d', teR Show that C'(1) = M'(1)e + M(1)e – é (b) Use (a) and (5.5) to show that C'(t) = 0 for all t e R. (c) Use Corollary 2 to show that C(t) is a constant and, hence, M(t) = C(0) = –1. (d) Show that (c) implies that M(t)e – é = -1, and therefore: M(t) = 1 – e-'.
57. Medication in the Human Body. We will solve a differential equation model for the concentration of drug in a patient's blood. Under a particular dosing regime, the amount M(t) in the pa- tient's blood changes in time according to the following differ- ential equation: = a – kịM, where a and ki are positive constants, representing the rate of absorption and the rate of elimination respectively, and the initial concentration is M(0) = 0. In Section 5.9 we will demonstrate that a solution to this equa- tion is M(t) = (1 – e-ki'). This question will show that this function is, in fact, the only solution of the equation. First let's as- sume that a = ki = 1 (this assumption simplifies the algebra, but the case of general parameters follows exactly the same lines). Assume that M(1) satisfies the differential equation: dM 31- М. М(0) — 0 dt (5.5)
Transcribed Image Text:57. Medication in the Human Body. We will solve a differential equation model for the concentration of drug in a patient's blood. Under a particular dosing regime, the amount M(t) in the pa- tient's blood changes in time according to the following differ- ential equation: = a – kịM, where a and ki are positive constants, representing the rate of absorption and the rate of elimination respectively, and the initial concentration is M(0) = 0. In Section 5.9 we will demonstrate that a solution to this equa- tion is M(t) = (1 – e-ki'). This question will show that this function is, in fact, the only solution of the equation. First let's as- sume that a = ki = 1 (this assumption simplifies the algebra, but the case of general parameters follows exactly the same lines). Assume that M(1) satisfies the differential equation: dM 31- М. М(0) — 0 dt (5.5)
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