Textbook Question
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
19. y = (sech θ)(1-ln(sech θ))
Ch. 7 - Transcendental Functions
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 7, Problem 7.2.88
Solve the initial value problems in Exercises 87 and 88.
88. d²y/dx² = sec²x, y(0)=0 and y'(0)=1
Verified step by step guidance1
Identify the given differential equation and initial conditions: \( \frac{d^2y}{dx^2} = \sec^2 x \), with \( y(0) = 0 \) and \( y'(0) = 1 \).
Integrate the second derivative \( \frac{d^2y}{dx^2} \) once with respect to \( x \) to find the first derivative \( y'(x) \). This means computing \( y'(x) = \int \sec^2 x \, dx + C_1 \), where \( C_1 \) is a constant of integration.
Use the initial condition \( y'(0) = 1 \) to solve for the constant \( C_1 \) after finding the integral expression for \( y'(x) \).
Integrate \( y'(x) \) with respect to \( x \) to find \( y(x) \). This involves computing \( y(x) = \int y'(x) \, dx + C_2 \), where \( C_2 \) is another constant of integration.
Apply the initial condition \( y(0) = 0 \) to solve for the constant \( C_2 \), completing the solution for \( y(x) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Second-Order Differential Equations
A second-order differential equation involves the second derivative of a function. Solving such equations typically requires integrating the given expression twice and applying initial conditions to find the constants of integration.
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Integration of Trigonometric Functions
Integrating functions like sec²x is essential in solving the differential equation. Knowing that the integral of sec²x is tan x helps in finding the first derivative, which can then be integrated again to find the original function.
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Initial Value Problems (IVP)
An initial value problem specifies values of the function and its derivatives at a particular point. These conditions allow us to determine the constants after integration, ensuring a unique solution that fits the given initial values.
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Related Practice
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