Textbook Question
Use Euler’s method with dx = 0.2 to estimate y(2) if y′ = y/x and y(1) = 2. What is the exact value of y(2)?
Ch. 9 - First-Order Differential Equations
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 9, Problem 9.2.24
Is either of the following equations correct? Give reasons for your answers.
a. (1/cosx) ∫ cos x dx = tan x + C
b. (1/cosx) ∫ cos x dx = tan x + C / cos x
Verified step by step guidance1
Start by evaluating the integral \( \int \cos x \, dx \). Recall that the integral of \( \cos x \) is \( \sin x + C \), where \( C \) is the constant of integration.
Rewrite the left side of both equations by substituting the integral: \( \frac{1}{\cos x} \int \cos x \, dx = \frac{1}{\cos x} (\sin x + C) \). This simplifies to \( \frac{\sin x}{\cos x} + \frac{C}{\cos x} \).
Recognize that \( \frac{\sin x}{\cos x} = \tan x \), so the expression becomes \( \tan x + \frac{C}{\cos x} \).
Compare this result with the right sides of the given equations: (a) \( \tan x + C \) and (b) \( \tan x + \frac{C}{\cos x} \). Notice that the constant term in (a) does not match the derived expression, while (b) matches exactly.
Conclude that equation (b) is correct because it properly accounts for the constant of integration divided by \( \cos x \), whereas equation (a) incorrectly treats the constant as a simple additive constant without division by \( \cos x \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration of Basic Trigonometric Functions
Understanding how to integrate basic trigonometric functions like cos x is essential. The integral of cos x with respect to x is sin x plus a constant of integration, C. This fundamental fact helps evaluate the given expressions correctly.
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Introduction to Trigonometric Functions
Algebraic Manipulation of Expressions
After integrating, it is important to correctly manipulate algebraic expressions, especially when dividing by functions like cos x. This includes distributing division over sums and understanding how constants of integration behave under division.
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6:36Simplifying Trig Expressions
Properties of the Constant of Integration
The constant of integration, C, represents an arbitrary constant and must be handled carefully. When dividing an integral expression by a function, the constant remains arbitrary and should not be combined or altered incorrectly, as it represents an infinite family of antiderivatives.
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06:21Properties of Functions
Related Practice
Textbook Question
Sailing A sailboat is running along a straight course with the wind providing a constant forward force of 50 lb. The only other force acting on the boat is resistance as the boat moves through the water. The resisting force is numerically equal to five times the boat’s speed, and the initial velocity is 1 ft/sec. What is the maximum velocity in feet per second of the boat under this wind?
Textbook Question
Solving Initial Value Problems
Solve the initial value problems in Exercises 15–20.
dy/dx + xy = x, y(0) = -6
Textbook Question
Using Euler’s Method
In Exercises 15–20, use Euler’s method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places.
y' = y²(1+2x), (y-1) = 1, dx = 0.5
Textbook Question
First-Order Linear Equations
Solve the differential equations in Exercises 1–14.
e²ˣy' + 2e²ˣ y = 2x
Textbook Question
In Exercises 39–42, use Euler’s method with the specified step size to estimate the value of the solution at the given point x*. Find the value of the exact solution at x*.
y' = 1 + y², y(0) = 0, dx = 0.1, x* = 1
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