Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 9 - Differential Equations

Problem 9.1.52b

Chapter 9, Problem 9.1.52b

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

{Use of Tech} Drug infusion The delivery of a drug (such as an antibiotic) through an intravenous line may be modeled by the differential equation m'(t) + km(t) = I, where m(t) is the mass of the drug in the blood at time t ≥ 0, k is a constant that describes the rate at which the drug is absorbed, and I is the infusion rate.

b. Graph the solution for I = 10 mg/hr and k = 0.05 hr⁻¹.

Verified step by step guidance

Identify the given differential equation: $m'(t) + k m(t) = I$, where $I = 10$ mg/hr and $k = 0.05$ hr$^{-1}$.

Recognize that this is a first-order linear ordinary differential equation. The standard approach is to find an integrating factor to solve it.

Calculate the integrating factor $\mu(t) = e^{\int k \, dt} = e^{k t}$, which in this case is $e^{0.05 t}$.

Multiply both sides of the differential equation by the integrating factor to get $e^{0.05 t} m'(t) + 0.05 e^{0.05 t} m(t) = 10 e^{0.05 t}$, which simplifies to $\frac{d}{dt} \left( e^{0.05 t} m(t) \right) = 10 e^{0.05 t}$.

Integrate both sides with respect to $t$ to find $e^{0.05 t} m(t) = \int 10 e^{0.05 t} dt + C$, then solve for $m(t)$ by dividing both sides by $e^{0.05 t}$. This will give the general solution for $m(t)$, which you can then graph.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

First-Order Linear Differential Equations

These are differential equations of the form m'(t) + p(t)m(t) = q(t), where the derivative of the function and the function itself appear linearly. The given drug infusion model fits this form, allowing the use of standard methods like integrating factors to find explicit solutions.

Integrating Factor Method

This technique solves first-order linear differential equations by multiplying both sides by an integrating factor, typically e^(∫p(t)dt), which simplifies the equation into an exact derivative. Applying this method to m'(t) + km(t) = I helps find the general solution for the drug mass over time.

Interpretation and Graphing of Solutions

Once the solution m(t) is found, understanding its behavior over time is crucial. For the drug infusion model, the solution typically approaches a steady-state value, reflecting equilibrium between infusion and absorption. Graphing with given parameters (I = 10, k = 0.05) visualizes this dynamic.

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Related Practice

Textbook Question

29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.

b. Using the exact solution given, compute the errors in the Euler approximations at t=0.2 and t=0.4.

y′(t) = −y, y(0) = 1; y(t) = e⁻ᵗ

Textbook Question

42–43. Implicit solutions for separable equations For the following separable equations, carry out the indicated analysis.

b. Find the value of the arbitrary constant associated with each initial condition. (Each initial condition requires a different constant.)

y'(t) = t²/(y² + 1); y(−1) = 1, y(0) = 0, y(−1) = −1

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Textbook Question

{Use of Tech} Intravenous drug dosing The amount of drug in the blood of a patient (in milligrams) administered via an intravenous line is governed by the initial value problem y’(t) = -0.02y + 3, y(0) = 0 where t is measured in hours.

b. What is the steady-state level of the drug?

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Textbook Question

A bad loan Consider a loan repayment plan described by the initial value problem

B'(t)=0.03B−600,B(0)=40,000,

where the amount borrowed is B(0)=\$40,000, the monthly payments are \$600, and B(t) is the unpaid balance in the loan.

b. What is the most that you can borrow under the terms of this loan without going further into debt each month?

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Textbook Question

{Use of Tech} Chemical rate equations Let y(t) be t he concentration of a substance in a chemical reaction (typical units are moles/liter). The change in the concentration, under appropriate conditions, is modeled by the equation dy/dt=-ky^n for t≥0, where k>0 is a rate constant and the positive integer n is the order of the reaction.

b. Solve the initial value problem for a second-order reaction (n=2) assuming y(0)=y0.

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Textbook Question

{Use of Tech} Torricelli’s law An open cylindrical tank initially filled with water drains through a hole in the bottom of the tank according to Torricelli’s law (see figure). If h(t) is the depth of water in the tank for t≥0 s, then Torricelli’s law implies h′(t)=−k√h, where k is a constant that includes g=9.8m/s², the radius of the tank, and the radius of the drain. Assume the initial depth of the water is h(0)=Hm.

b. Find the solution in k=0.1the case that and H=0.5m.

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