Skip to main content
Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 9 - Differential EquationsProblem 9.3.41c
Chapter 9, Problem 9.3.41c

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. The general solution of the equation yy'(x) = xe⁻ʸ can be found using integration by parts.

Verified step by step guidance
1
First, rewrite the given differential equation in a more explicit form: \(y \cdot y'(x) = x e^{-y}\), where \(y'(x) = \frac{dy}{dx}\).
Recognize that the equation involves both \(y\) and its derivative \(y'\), and the right side contains \(x\) and an exponential function of \(-y\).
Check if the equation can be separated into variables, i.e., express it as a product of a function of \(y\) and a function of \(x\) to integrate both sides separately.
Integration by parts is a technique used to integrate products of functions with respect to a single variable, typically in the form \(\int u \, dv = uv - \int v \, du\). Since this is a differential equation involving \(y\) and \(x\), integration by parts is not directly applicable to solve it.
Instead, try to separate variables or use substitution methods to solve the equation. For example, rewrite as \(y \frac{dy}{dx} = x e^{-y}\), then rearrange to \(y e^{y} dy = x dx\) if possible, and integrate both sides accordingly.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
Was this helpful?

Key Concepts

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

Differential Equations and General Solutions

A differential equation relates a function and its derivatives. The general solution includes all possible solutions and often contains arbitrary constants. Understanding how to find the general solution is essential to determine if a particular method applies.
Recommended video:
04:00
Solutions to Basic Differential Equations

Integration by Parts

Integration by parts is a technique derived from the product rule for differentiation, used to integrate products of functions. It transforms an integral into simpler parts but is typically applied to integrals, not directly to differential equations.
Recommended video:
06:18
Integration by Parts for Definite Integrals

Methods for Solving First-Order Differential Equations

First-order differential equations can be solved using methods like separation of variables, integrating factors, or substitution. Recognizing the equation type helps decide the appropriate method, and integration by parts is generally not a direct method for solving such equations.
Recommended video:
06:06
Solving Separable Differential Equations
Related Practice
Textbook Question

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.

c. Repeating parts (a) and (b) using half the time step used in those calculations, again find an approximation to y(T).


y′(t) = 6 - 2y, y(0) = -1; Δt = 0.2, T = 3; y(t) = 3 - 4e⁻²ᵗ

1
views
Textbook Question

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.

b. Using the exact solution (also given), find the error in the approximation to y(T) (only at the right endpoint of the time interval).


y′(t) = t/y, y(0) = 4; Δt = 0.1, T = 2; y(t) = √(t² + 16)

1
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) Chemical rate equations The reaction of certain chemical compounds can be modeled using a differential equation of the form y'(t) = -kyⁿ(t), where y(t) is the concentration of the compound, for t ≥ 0, k > 0 is a constant that determines the speed of the reaction, and n is a positive integer called the order of the reaction. Assume the initial concentration of the compound is y(0) = y₀ > 0.


c. Let y₀ = 1 and k = 0.1. Graph the first-order and second-order solutions found in parts (a) and (b). Compare the two reactions. 

Textbook Question

{Use of Tech} Free fall An object in free fall may be modeled by assuming the only forces at work are the gravitational force and air resistance. By Newton’s Second Law of Motion (mass end . acceleration = the sum of external forces), the velocity of the object satisfies the differential equation 


m · v'(t) = mg + f(v)

mass | acceleration | external forces


where f is a function that models the air resistance (assuming the positive direction is downward). One common assumption (often used for motion in air) is that f(v)=−kv^2, for t≥0, where k>0 is a drag coefficient.


c. Find the solution of this separable equation assuming v(0)=0 and 0<v²<g/a. 

1
views
Textbook Question

Direction field analysis Consider the first-order initial value problem y'(t)=ay+b,y(0)=A for t≥0 where a, b, and A are real numbers.

c. Draw a representative direction field in the case that a<0. Show that if A>−b/a, then the solution decreases for t≥0, and that if A<−b/a, then the solution increases for t≥0.

1
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} Free fall One possible model that describes the free fall of an object in a gravitational field subject to air resistance uses the equation v'(t) = g - bv, where v(t) is the velocity of the object for t ≥ 0, g = 9.8 m/s² is the acceleration due to gravity, and b > 0 is a constant that involves the mass of the object and the air resistance.


c. Using the graph in part (b), estimate the terminal velocity lim(t→∞) v(t).