Use the method for solving homogeneous equations to solve the following differential equation. 4(x² + y²) dx + 5xy dy=0
Use the method for solving homogeneous equations to solve the following differential equation. 4(x² + y²) dx + 5xy dy=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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![### Solving Homogeneous Differential Equations
**Problem Statement:**
Use the method for solving homogeneous equations to solve the following differential equation:
\[ 4 \left( \frac{x^2 + y^2}{x} \right) dx + 5xy \, dy = 0 \]
**Solution Instructions:**
1. **Understanding the Differential Equation:**
The given differential equation is:
\[ 4 \left( \frac{x^2 + y^2}{x} \right) dx + 5xy \, dy = 0 \]
2. **Objective:**
Ignoring lost solutions, if any, an implicit solution in the form \( F(x,y) = C \) can be found. Here, \( C \) is an arbitrary constant.
3. **Formulating the Solution:**
Find the implicit solution using \( x \) and \( y \) as the variables. Type the expression in the provided form to ensure correctness.
**Final Expression:**
\[ \boxed{ \ } = C \]
**Instructions for Completing the Expression:**
Type an expression using \( x \) and \( y \) as the variables to complete the boxed equation. The boxed area represents the place where the implicit solution should be entered.
**Note:**
Be sure to follow the principles of solving homogeneous differential equations to arrive at the correct form of the implicit solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd3485286-e420-456a-8a41-5bc62e18ff5f%2F6f34bd3a-299a-4dcc-88cd-da196192420a%2F0v4qrq_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Homogeneous Differential Equations
**Problem Statement:**
Use the method for solving homogeneous equations to solve the following differential equation:
\[ 4 \left( \frac{x^2 + y^2}{x} \right) dx + 5xy \, dy = 0 \]
**Solution Instructions:**
1. **Understanding the Differential Equation:**
The given differential equation is:
\[ 4 \left( \frac{x^2 + y^2}{x} \right) dx + 5xy \, dy = 0 \]
2. **Objective:**
Ignoring lost solutions, if any, an implicit solution in the form \( F(x,y) = C \) can be found. Here, \( C \) is an arbitrary constant.
3. **Formulating the Solution:**
Find the implicit solution using \( x \) and \( y \) as the variables. Type the expression in the provided form to ensure correctness.
**Final Expression:**
\[ \boxed{ \ } = C \]
**Instructions for Completing the Expression:**
Type an expression using \( x \) and \( y \) as the variables to complete the boxed equation. The boxed area represents the place where the implicit solution should be entered.
**Note:**
Be sure to follow the principles of solving homogeneous differential equations to arrive at the correct form of the implicit solution.
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