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For the following problems, set up and solve the differential equations.A drug is administered to a patient every 24 hours and is cleared at a rate proportional to the amount of drug left in the body, with proportionality constant $0.2 .$ If the patient needs a baseline level of 5 $\mathrm{mg}$ to be in the bloodstream at all times, how large should the dose be?

$y=\frac{r}{d}\left(1-e^{-d t}\right)$

Calculus 2 / BC

Chapter 4

Introduction to Differential Equations

Section 5

First-order Linear Equations

Differential Equations

Missouri State University

Harvey Mudd College

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Lectures

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Intravenous drug dosing Th…

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Okay, So what we have here is a problem involving a drug which has arrayed proportional to the amount of drug in the blood stream at any time. P So we are also given that a physical 20 Q is equal to a few knots and the three piece equals three que easy quota point when you're not talking. So we are required to get you off B intersil for the among off drug in the blood stream at peace equal to six if you're not a sequel to 100 milligram. So for our solution, we can represent you over. The V is equal. Do now I got you King fue where in K is the proportionality constant So integrating this will have Vic you over Q physical toe the negative gave BP and getting the integral of both sides will have. And in a few music was the negative. Katie, let's see. So today, seeing um, getting the power of the of both sides will have e off. Ellen of you is equal toe. Erase the negative. Katie, Let's see. So remember here that sees disa call center, it can take up any value and our final and are resulting equation is Q is equal to see here days to the negative, baby. So this is our general solution. So we want to find the body off, See, nt and we do this making use of the initial conditions even in the problem. So AT T is equal to zero. It was said that you is equal to appear not so you're not is a well toe See theories in the negative k, thank zero. So this will just cancel out the one therefore C is equal. Look, you're not a resulting equation. Then Sq is equal to you're not theories to the negative, Katie. So now we want to find the body off key. So a physical good three. We know that Q is equal to the point. Wendy off. You're not to substituting this over equation will have point when you're not, is equal to fear not. Ian is the negative gay 93 to be suggest Cancel out and we are left with 0.20 sequel To erase to the negative three K and solving Porky, we would have zero point 54. So reviving the Holik we should que off b is equal to. You're not e eight days in the negative 0.54 feet. So we'll use these to solve the problem in that beat for leather be. We have the sick with the seats and you're not physical to 100 meaning it up. So computing for Q Off six, we have 100 erased in the negative 0.54 times six, which is approximately equal to 3.92 Milica. So this is our final answer.

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