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 9.3.50b

Chapter 9, Problem 9.3.50b

{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.

Verified step by step guidance

Identify the given differential equation for the second-order reaction: $\frac{dy}{dt} = -k y^{2}$, where $k > 0$ and the initial condition is $y(0) = y_0$.

Rewrite the differential equation to separate variables, moving all terms involving $y$ to one side and $t$ to the other: $\frac{dy}{y^{2}} = -k \, dt$.

Integrate both sides: integrate $\int \frac{dy}{y^{2}}$ with respect to $y$ and $\int -k \, dt$ with respect to $t$.

After integration, apply the initial condition $y(0) = y_0$ to solve for the constant of integration.

Finally, solve the resulting equation for $y$ explicitly as a function of $t$ to express the concentration $y(t)$.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Separable Differential Equations

A separable differential equation can be written as dy/dt = g(t)h(y), allowing variables y and t to be separated on opposite sides. This enables integration with respect to each variable independently, which is essential for solving equations like dy/dt = -ky^n.

Initial Value Problem (IVP)

An initial value problem specifies the value of the unknown function at a particular point, such as y(0) = y0. This condition allows us to find the particular solution to a differential equation that fits the given starting concentration.

Second-Order Reaction Rate Law

For a second-order reaction (n=2), the rate equation is dy/dt = -ky^2. Solving this involves integrating the reciprocal of y squared, leading to a solution that describes how concentration decreases over time, typically resulting in a rational function of t.

Recommended video:

06:35

Changing Geometries

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

views

Textbook Question

38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

b. Sketch the direction field, for t≥0.

y′(t) = 6 - 2y

views

Textbook Question

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.

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?

views

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.

views