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In this problem we consider a patient being given a drug through infusion, for example through the steady drip of an IV, instead of one pill at a time. If m(t) is the mass of the drug in the blood at time t≥ 0, R is a constant infusion rate, and k> 0 is a constant rate at which the drug is absorbed from the blood into the body, the corresponding differential equation is dm dt = R-k-m(t). R(1-e For the initial condition m(0) - e-kt) = 0 (no drug in the blood initially), the solution is m(t) k Notice that the solution depends on time (obviously), but also on the constants R and k. Hint: only one of these three questions requires L'Hopital's Rule. - 1. What is the long-term behavior of the solution (that is, time goes to infinity)? Why does the answer make sense? 2. What does the solution look like for an infinitely large absorption rate k? (Compute the limit of m(t) as k→ ∞.) Why does the answer make sense? 3. What does the solution look like when the absorption rate k goes to 0? (Compute the limit of m(t) as k→ 0+.) Why does this answer make sense?

In this problem we consider a patient being given a drug through infusion, for example through the steady drip of an IV, instead of one pill at a time. If m(t) is the mass of the drug in the blood at time t≥ 0, R is a constant infusion rate, and k> 0 is a constant rate at which the drug is absorbed from the blood into the body, the corresponding differential equation is dm dt = R-k-m(t). R(1-e For the initial condition m(0) - e-kt) = 0 (no drug in the blood initially), the solution is m(t) k Notice that the solution depends on time (obviously), but also on the constants R and k. Hint: only one of these three questions requires L'Hopital's Rule. - 1. What is the long-term behavior of the solution (that is, time goes to infinity)? Why does the answer make sense? 2. What does the solution look like for an infinitely large absorption rate k? (Compute the limit of m(t) as k→ ∞.) Why does the answer make sense? 3. What does the solution look like when the absorption rate k goes to 0? (Compute the limit of m(t) as k→ 0+.) Why does this answer make sense?

Advanced Engineering Mathematics
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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In this problem we consider a patient being given a drug through infusion, for example through the steady drip of an IV, instead of one pill at a time. If m(t) is the mass of the drug in the blood at time t≥ 0, R is a constant infusion rate, and k> 0 is a constant rate at which the drug is absorbed from the blood into the body, the corresponding differential equation is dm dt = R-k m(t). e-kt) R(1 - e k For the initial condition m(0) = 0 (no drug in the blood initially), the solution is m(t) Notice that the solution depends on time (obviously), but also on the constants R and k. Hint: only one of these three questions requires L'Hopital's Rule. 1. What is the long-term behavior of the solution (that is, time goes to infinity)? Why does the answer make sense? 2. What does the solution look like for an infinitely large absorption rate k? (Compute the limit of m(t) as k→ ∞.) Why does the answer make sense? 3. What does the solution look like when the absorption rate k goes to 0? (Compute the limit of m(t) as k→ 0+.) Why does this answer make sense?
Transcribed Image Text:In this problem we consider a patient being given a drug through infusion, for example through the steady drip of an IV, instead of one pill at a time. If m(t) is the mass of the drug in the blood at time t≥ 0, R is a constant infusion rate, and k> 0 is a constant rate at which the drug is absorbed from the blood into the body, the corresponding differential equation is dm dt = R-k m(t). e-kt) R(1 - e k For the initial condition m(0) = 0 (no drug in the blood initially), the solution is m(t) Notice that the solution depends on time (obviously), but also on the constants R and k. Hint: only one of these three questions requires L'Hopital's Rule. 1. What is the long-term behavior of the solution (that is, time goes to infinity)? Why does the answer make sense? 2. What does the solution look like for an infinitely large absorption rate k? (Compute the limit of m(t) as k→ ∞.) Why does the answer make sense? 3. What does the solution look like when the absorption rate k goes to 0? (Compute the limit of m(t) as k→ 0+.) Why does this answer make sense?
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