the explicit solution to the differential equation (2y² + 4x²) dx-xy dy=0; y(1) = -2
the explicit solution to the differential equation (2y² + 4x²) dx-xy dy=0; y(1) = -2
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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Question
![**Differential Equations - Solving Techniques**
**Problem Statement:**
Find the **explicit** solution to the differential equation:
\[
(2y^2 + 4x^2) \, dx - xy \, dy = 0 \quad ; \quad y(1) = -2
\]
**Explanation:**
This problem provides a differential equation involving variables \(x\) and \(y\) and requires finding an explicit solution that satisfies the given initial condition \(y(1) = -2\).
**Steps to Approach:**
1. **Identify the type of differential equation.**
2. **Separate the variables** if possible.
3. **Integrate both sides** to find the general solution.
4. **Apply the initial condition** to determine the constant of integration.
5. **Write the explicit solution.**
Understanding these steps will aid in solving similar types of differential equations and reinforce knowledge on integrating factors, separation of variables, or other relevant methods.
**Graphical Representation:**
Currently, there are **no graphs or diagrams** provided with this problem. However, visualizing such solutions often involves plotting the function \(y(x)\) once the explicit solution is found, especially to understand the behavior of the function across different \(x\) values and ensure it meets the given initial conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F946e5b6f-098d-47d1-a6b1-15872ebd1054%2F9aa749f6-e710-480a-a5df-3e751c1a419a%2Fh80mt2_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Differential Equations - Solving Techniques**
**Problem Statement:**
Find the **explicit** solution to the differential equation:
\[
(2y^2 + 4x^2) \, dx - xy \, dy = 0 \quad ; \quad y(1) = -2
\]
**Explanation:**
This problem provides a differential equation involving variables \(x\) and \(y\) and requires finding an explicit solution that satisfies the given initial condition \(y(1) = -2\).
**Steps to Approach:**
1. **Identify the type of differential equation.**
2. **Separate the variables** if possible.
3. **Integrate both sides** to find the general solution.
4. **Apply the initial condition** to determine the constant of integration.
5. **Write the explicit solution.**
Understanding these steps will aid in solving similar types of differential equations and reinforce knowledge on integrating factors, separation of variables, or other relevant methods.
**Graphical Representation:**
Currently, there are **no graphs or diagrams** provided with this problem. However, visualizing such solutions often involves plotting the function \(y(x)\) once the explicit solution is found, especially to understand the behavior of the function across different \(x\) values and ensure it meets the given initial conditions.
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