Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Initial-Value Problem
Solve the initial-value problem given below:
\[ x' + 7x = e^{-7t} \cos(t), \quad x(0) = -1 \]
`x(t) =` [Input box]
\[\large \mathbf{\xmark}\]
#### Explanation:
The equation provided is a first-order linear differential equation with initial condition. Follow these steps to find the solution for \( x(t) \):
1. **Identify the standard form** of the linear differential equation:
\[ x' + P(t)x = Q(t) \]
Here, \( P(t) = 7 \) and \( Q(t) = e^{-7t} \cos(t) \).
2. **Determine the integrating factor**:
\[ \mu(t) = e^{\int P(t) \, dt} = e^{7t} \]
3. **Multiply through by the integrating factor** and solve:
\[ e^{7t} x' + 7e^{7t} x = e^{7t} e^{-7t} \cos(t) \]
This simplifies to:
\[ \frac{d}{dt}\left(e^{7t} x \right) = \cos(t) \]
4. **Integrate both sides**:
\[ e^{7t} x = \int \cos(t) \, dt = \sin(t) + C \]
5. **Solve for \( x(t) \)**:
\[ x = e^{-7t} (\sin(t) + C) \]
6. **Apply the initial condition \( x(0) = -1 \)**:
\[ -1 = e^{0} (\sin(0) + C) \]
\[ -1 = C \]
So, the solution is:
\[ x(t) = e^{-7t} (\sin(t) - 1) \]
Plugging this back in the input box on the website should give you the correct result.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F004f485a-dc8e-4d2a-a997-494cbc4bff99%2F170d9854-28f0-415e-a839-cbd51a5ed38f%2Fy7i5d9nl_processed.png&w=3840&q=75)
Transcribed Image Text:### Initial-Value Problem
Solve the initial-value problem given below:
\[ x' + 7x = e^{-7t} \cos(t), \quad x(0) = -1 \]
`x(t) =` [Input box]
\[\large \mathbf{\xmark}\]
#### Explanation:
The equation provided is a first-order linear differential equation with initial condition. Follow these steps to find the solution for \( x(t) \):
1. **Identify the standard form** of the linear differential equation:
\[ x' + P(t)x = Q(t) \]
Here, \( P(t) = 7 \) and \( Q(t) = e^{-7t} \cos(t) \).
2. **Determine the integrating factor**:
\[ \mu(t) = e^{\int P(t) \, dt} = e^{7t} \]
3. **Multiply through by the integrating factor** and solve:
\[ e^{7t} x' + 7e^{7t} x = e^{7t} e^{-7t} \cos(t) \]
This simplifies to:
\[ \frac{d}{dt}\left(e^{7t} x \right) = \cos(t) \]
4. **Integrate both sides**:
\[ e^{7t} x = \int \cos(t) \, dt = \sin(t) + C \]
5. **Solve for \( x(t) \)**:
\[ x = e^{-7t} (\sin(t) + C) \]
6. **Apply the initial condition \( x(0) = -1 \)**:
\[ -1 = e^{0} (\sin(0) + C) \]
\[ -1 = C \]
So, the solution is:
\[ x(t) = e^{-7t} (\sin(t) - 1) \]
Plugging this back in the input box on the website should give you the correct result.
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