4. Insulin Pump An insulin pump is used to treat diabetes by continuously infusing insulin into the fat in a patient's abdomen at a constant rate r = 2.From there a fraction f = .90 of the insulin in the fat diffuses into the patients blood in each unit time. If x(t) is the amount of insulin in the patient's fat,then x(t) is a solution to the differential equation:%3Dda=r - fr.dt(a) Find the general solution to this differential equation. Be sure to discuss what kind of differential equation this is and what steps you took to solve it.(b) Assuming that ¤(0) = 0, find x(t). Using your answer, find the long-term behavior of r(t) (in other words, find lim- T(t)). Interpret your answer.

Here, r=2f=0.90a) Substitute the values into the given equation. dxdt=2-0.9xdx2-0.9x=dt So, given differential equation is a type of separable differential equation. Integrate both sides. ∫dx2-0.9x=∫dtln2-0.9x-0.9=t+cln2-0.9x=-0.9t+c12-0.9x=c2e-0.9t0.9x=2-c2e-0.9tx=2.22-Ce-0.9tb) Find the value of C using the condition, x0=0 0=2.22-Ce-0.9×02.22=C Substitute the value of C into the general solution. x=2.22-2.22e-0.9t=2.221-e-0.9tFind the limit when t approaches to ∞. limt→∞xt=limt→∞2.221-e-0.9t=2.221-e-0.9×∞=2.221-1∞=2.22 So, in long term, the amount of insulin left in patients' fat is 2.22

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. Insulin Pump An insulin pump is used to treat diabetes by continuously infusing insulin into the fat in a patient's abdomen at a constant rater = 2. From there a fraction ƒ = .90 of the insulin in the fat diffuses into the patients blood in each unit time. If r(t) is the amount of insulin in the patient's fat, then r(t) is a solution to the differential equation: dz =r- fr. dt (a) Find the general solution to this differential equation. Be sure to discuss what kind of differential equation this is and what steps you took to solve it. (b) Assuming that ¤(0) = 0, find æ(t). Using your answer, find the long-term behavior of r(t) (in other words, find lim-+∞ ¤(t)). Interpret your answer.

. Insulin Pump An insulin pump is used to treat diabetes by continuously infusing insulin into the fat in a patient's abdomen at a constant rater = 2. From there a fraction ƒ = .90 of the insulin in the fat diffuses into the patients blood in each unit time. If r(t) is the amount of insulin in the patient's fat, then r(t) is a solution to the differential equation: dz =r- fr. dt (a) Find the general solution to this differential equation. Be sure to discuss what kind of differential equation this is and what steps you took to solve it. (b) Assuming that ¤(0) = 0, find æ(t). Using your answer, find the long-term behavior of r(t) (in other words, find lim-+∞ ¤(t)). Interpret your answer.

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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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.

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4. Insulin Pump An insulin pump is used to treat diabetes by continuously infusing insulin into the fat in a patient's abdomen at a constant rate r = 2.
From there a fraction f = .90 of the insulin in the fat diffuses into the patients blood in each unit time. If x(t) is the amount of insulin in the patient's fat,
then x(t) is a solution to the differential equation:
%3D
da
=r - fr.
dt
(a) Find the general solution to this differential equation. Be sure to discuss what kind of differential equation this is and what steps you took to solve it.
(b) Assuming that ¤(0) = 0, find x(t). Using your answer, find the long-term behavior of r(t) (in other words, find lim- T(t)). Interpret your answer.

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