y' - ycosx = cos x
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Understanding First-Order Linear Differential Equations**
On this educational page, we discuss first-order linear differential equations. The differential equation presented here is:
\[ y' - y \cos x = \cos x \]
### Explanation of Terms:
- \( y' \) denotes the first derivative of \( y \) with respect to \( x \).
- \( \cos x \) is the cosine function, which is a trigonometric function of \( x \).
### Structure of the Equation:
This equation is of the form:
\[ y' + P(x)y = Q(x) \]
where:
- \( P(x) = -\cos x \)
- \( Q(x) = \cos x \)
### Steps to Solve the Equation:
1. **Identify** \( P(x) \) and \( Q(x) \).
2. **Find the integrating factor** \( \mu(x) \), which is given by:
\[ \mu(x) = e^{\int P(x) \, dx} \]
3. **Multiply** the whole differential equation by \( \mu(x) \) to facilitate finding the solution.
### Detailed Steps:
1. For our equation:
\[ P(x) = -\cos x \]
2. The integral of \( P(x) \):
\[ \int -\cos x \, dx = -\sin x \]
3. The integrating factor:
\[ \mu(x) = e^{-\sin x} \]
4. Multiplying the original equation by \( \mu(x) \):
\[ e^{-\sin x} y' - e^{-\sin x} y \cos x = e^{-\sin x} \cos x \]
which simplifies the integration process.
### Final Step:
After finding the integrating factor, you integrate both sides with respect to \( x \) to find the general solution for \( y \).
### Additional Notes:
For visual learners, it may be beneficial to graph the functions involved using graphing software or tools. Understanding how \( \cos x \) behaves helps intuitively grasp how \( y \) changes with respect to \( x \).
Feel free to experiment by solving similar differential equations to solidify your understanding!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6aad8a00-c703-485c-bf76-99694ae31b4b%2F88523fe5-d35d-4c4b-9bbb-4be5a6cde30a%2Fv91nkn6_processed.png&w=3840&q=75)
Transcribed Image Text:**Understanding First-Order Linear Differential Equations**
On this educational page, we discuss first-order linear differential equations. The differential equation presented here is:
\[ y' - y \cos x = \cos x \]
### Explanation of Terms:
- \( y' \) denotes the first derivative of \( y \) with respect to \( x \).
- \( \cos x \) is the cosine function, which is a trigonometric function of \( x \).
### Structure of the Equation:
This equation is of the form:
\[ y' + P(x)y = Q(x) \]
where:
- \( P(x) = -\cos x \)
- \( Q(x) = \cos x \)
### Steps to Solve the Equation:
1. **Identify** \( P(x) \) and \( Q(x) \).
2. **Find the integrating factor** \( \mu(x) \), which is given by:
\[ \mu(x) = e^{\int P(x) \, dx} \]
3. **Multiply** the whole differential equation by \( \mu(x) \) to facilitate finding the solution.
### Detailed Steps:
1. For our equation:
\[ P(x) = -\cos x \]
2. The integral of \( P(x) \):
\[ \int -\cos x \, dx = -\sin x \]
3. The integrating factor:
\[ \mu(x) = e^{-\sin x} \]
4. Multiplying the original equation by \( \mu(x) \):
\[ e^{-\sin x} y' - e^{-\sin x} y \cos x = e^{-\sin x} \cos x \]
which simplifies the integration process.
### Final Step:
After finding the integrating factor, you integrate both sides with respect to \( x \) to find the general solution for \( y \).
### Additional Notes:
For visual learners, it may be beneficial to graph the functions involved using graphing software or tools. Understanding how \( \cos x \) behaves helps intuitively grasp how \( y \) changes with respect to \( x \).
Feel free to experiment by solving similar differential equations to solidify your understanding!
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