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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)?
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Ch. 9 - First-Order Differential Equations
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 9, Problema 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
Guida verificata passo dopo passo1
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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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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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
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Solve the initial value problems in Exercises 15–20.
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First-Order Linear Equations
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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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