dv dx X- 1-4v² Jv² Bv
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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---
### Differential Equation Problem
#### Problem Statement:
\[ \frac{x}{dx} \frac{dv}{dx} = \frac{1 - 4v^2}{3v} \]
In the given problem, we have a first-order differential equation that needs to be solved. The equation can be interpreted as a separable differential equation, which may be solved by algebraic manipulation and integration.
#### Detailed Breakdown:
1. Multiply both sides of the equation by \(dx\):
\[ x \, dv = \frac{(1 - 4v^2) dx}{3v} \]
2. Separate the variables \(v\) and \(x\):
\[ 3v \, dv = \frac{1 - 4v^2}{x} \, dx \]
3. Integrate both sides to find the general solution.
By solving this differential equation, we can find the function \(v(x)\) that satisfies this relation.
#### Graph/Diagram Explanation:
There is a red curve drawn on the right side of the image. The curve might represent an example of a solution to the differential equation, showing the relationship between variables \(v\) and \(x\). However, to accurately determine the exact nature of the curve, further context or specific solutions must be calculated.
---](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F13675a72-9a5d-4ab9-ac06-e7f5015e8566%2Fab423b10-4138-4cfc-89b4-a3266489be77%2Fzxx51e7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! Below is the transcribed text suitable for an educational website:
---
### Differential Equation Problem
#### Problem Statement:
\[ \frac{x}{dx} \frac{dv}{dx} = \frac{1 - 4v^2}{3v} \]
In the given problem, we have a first-order differential equation that needs to be solved. The equation can be interpreted as a separable differential equation, which may be solved by algebraic manipulation and integration.
#### Detailed Breakdown:
1. Multiply both sides of the equation by \(dx\):
\[ x \, dv = \frac{(1 - 4v^2) dx}{3v} \]
2. Separate the variables \(v\) and \(x\):
\[ 3v \, dv = \frac{1 - 4v^2}{x} \, dx \]
3. Integrate both sides to find the general solution.
By solving this differential equation, we can find the function \(v(x)\) that satisfies this relation.
#### Graph/Diagram Explanation:
There is a red curve drawn on the right side of the image. The curve might represent an example of a solution to the differential equation, showing the relationship between variables \(v\) and \(x\). However, to accurately determine the exact nature of the curve, further context or specific solutions must be calculated.
---
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