dr ᎾᏢ sin 0 - r- cos e eo +- cos = 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![### Example Problem 2
Solve the differential equation:
\[
\frac{dr}{d\theta} - r \frac{\sin \theta}{\cos \theta} + \frac{e^\theta}{\cos \theta} = 0
\]
### Explanation
In this differential equation, we are given:
- \( \frac{dr}{d\theta} \): The derivative of \( r \) with respect to \( \theta \).
- \( \sin \theta \) and \( \cos \theta \): Trigonometric functions of \( \theta \).
- \( e^\theta \): The exponential function of \( \theta \).
The goal is to manipulate this equation to find a function \( r(\theta) \) that satisfies this differential equation. Starting with the given equation and simplifying or using methods like integrating factors might be necessary to solve it.
### Steps to Solve
1. Rearrange the equation to isolate terms involving \( r \) on one side if possible:
\[
\frac{dr}{d\theta} = r \frac{\sin \theta}{\cos \theta} - \frac{e^\theta}{\cos \theta}
\]
2. Recognize that the term \( \frac{\sin \theta}{\cos \theta} \) is \( \tan \theta \).
3. You will have:
\[
\frac{dr}{d\theta} = r \tan \theta - \frac{e^\theta}{\cos \theta}
\]
4. Solve this differential equation using appropriate mathematical techniques such as integrating factors or separation of variables.
Understanding how to manipulate and solve this type of differential equation is key in various applications in physics and engineering, especially those involving polar coordinates and trigonometric functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F13675a72-9a5d-4ab9-ac06-e7f5015e8566%2F1820e333-64ef-44a1-bc03-fef36a3315e7%2Faj6whos_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Example Problem 2
Solve the differential equation:
\[
\frac{dr}{d\theta} - r \frac{\sin \theta}{\cos \theta} + \frac{e^\theta}{\cos \theta} = 0
\]
### Explanation
In this differential equation, we are given:
- \( \frac{dr}{d\theta} \): The derivative of \( r \) with respect to \( \theta \).
- \( \sin \theta \) and \( \cos \theta \): Trigonometric functions of \( \theta \).
- \( e^\theta \): The exponential function of \( \theta \).
The goal is to manipulate this equation to find a function \( r(\theta) \) that satisfies this differential equation. Starting with the given equation and simplifying or using methods like integrating factors might be necessary to solve it.
### Steps to Solve
1. Rearrange the equation to isolate terms involving \( r \) on one side if possible:
\[
\frac{dr}{d\theta} = r \frac{\sin \theta}{\cos \theta} - \frac{e^\theta}{\cos \theta}
\]
2. Recognize that the term \( \frac{\sin \theta}{\cos \theta} \) is \( \tan \theta \).
3. You will have:
\[
\frac{dr}{d\theta} = r \tan \theta - \frac{e^\theta}{\cos \theta}
\]
4. Solve this differential equation using appropriate mathematical techniques such as integrating factors or separation of variables.
Understanding how to manipulate and solve this type of differential equation is key in various applications in physics and engineering, especially those involving polar coordinates and trigonometric functions.
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