Find the solution of the following differential equation. + 6.x ) dx + ( In x – 2 )dy = 0, x > 0.
Find the solution of the following differential equation. + 6.x ) dx + ( In x – 2 )dy = 0, x > 0.
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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![### Problem 6: Differential Equation Solution
Find the solution of the following differential equation:
\[
\left( \frac{y}{x} + 6x \right) dx + \left( \ln x - 2 \right) dy = 0, \quad x > 0.
\]
This problem involves solving a first order differential equation, where \(x > 0\). The equation is presented in the form of a linear combination of differential terms in \(dx\) and \(dy\).
**Explanation**
- **Expression:** The differential equation is composed of two parts:
- \(\left( \frac{y}{x} + 6x \right) dx\)
- \(\left( \ln x - 2 \right) dy\)
- **Condition:** The condition \(x > 0\) implies that the function is defined for positive \(x\).
This type of equation may be approached by methods such as separation of variables, integrating factor, or assuming a particular solution form, depending on the context and form of the differential equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5bbc1593-fa0b-47cc-b65c-3df8b16301c5%2F40d01515-8f03-4c15-b0cb-f0b5e5716322%2F0v2j3up_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 6: Differential Equation Solution
Find the solution of the following differential equation:
\[
\left( \frac{y}{x} + 6x \right) dx + \left( \ln x - 2 \right) dy = 0, \quad x > 0.
\]
This problem involves solving a first order differential equation, where \(x > 0\). The equation is presented in the form of a linear combination of differential terms in \(dx\) and \(dy\).
**Explanation**
- **Expression:** The differential equation is composed of two parts:
- \(\left( \frac{y}{x} + 6x \right) dx\)
- \(\left( \ln x - 2 \right) dy\)
- **Condition:** The condition \(x > 0\) implies that the function is defined for positive \(x\).
This type of equation may be approached by methods such as separation of variables, integrating factor, or assuming a particular solution form, depending on the context and form of the differential equation.
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