Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
83. y = 3^(log₂ t)
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 7.3.57
Solve the initial value problems in Exercises 55–58.
57. d²y/dx² = 2e^(−x),y(0) = 1,y′(0) = 0
Verified step by step guidance1
Identify the given differential equation and initial conditions: \( \frac{d^{2}y}{dx^{2}} = 2e^{-x} \), with \( y(0) = 1 \) and \( y'(0) = 0 \).
Integrate the second derivative \( \frac{d^{2}y}{dx^{2}} \) once with respect to \( x \) to find the first derivative \( y'(x) \). This gives \( y'(x) = \int 2e^{-x} \, dx + C_1 \), where \( C_1 \) is a constant of integration.
Perform the integration \( \int 2e^{-x} \, dx \) by recalling that \( \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \). Substitute the result back to express \( y'(x) \) including the constant \( C_1 \).
Integrate \( y'(x) \) with respect to \( x \) to find \( y(x) \). This will introduce another constant of integration \( C_2 \). So, \( y(x) = \int y'(x) \, dx = \int (\text{expression from previous step}) \, dx + C_2 \).
Use the initial conditions \( y(0) = 1 \) and \( y'(0) = 0 \) to set up equations and solve for the constants \( C_1 \) and \( C_2 \). Substitute these constants back into the expression for \( y(x) \) to get the particular solution.
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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 an unknown function. Solving it requires finding a function y(x) whose second derivative matches the given expression. The general solution often includes two arbitrary constants, reflecting the equation's order.
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Classifying Differential Equations
Initial Conditions and Initial Value Problems
Initial conditions specify the values of a function and its derivatives at a particular point, allowing determination of the arbitrary constants in the general solution. An initial value problem combines a differential equation with these conditions to find a unique solution.
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Integration of Exponential Functions
Solving the differential equation involves integrating the right-hand side function, here an exponential function e^(−x). Understanding how to integrate exponentials and apply constants of integration is essential to find the general solution before applying initial conditions.
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Related Practice
Textbook Question
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
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Use l’Hôpital’s rule to find the limits in Exercises 7–52.
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In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
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Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
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