Using separation of variables, solve the differential equation, Use C to represent the arbitrary constant. y² = dy da (5+x¹).
Using separation of variables, solve the differential equation, Use C to represent the arbitrary constant. y² = dy da (5+x¹).
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
![**Problem Statement: Solving a Differential Equation Using Separation of Variables**
**Task:**
Using the method of separation of variables, solve the differential equation provided below:
\[
(5 + x^4) \frac{dy}{dx} = \frac{x^3}{y}
\]
Make sure to use **C** to represent the arbitrary constant.
**Solution Format:**
\[ y^2 = \]
[Insert the solution derived from the separation of variables method here.]
**Explanation of Approach:**
To tackle this problem, follow these steps:
1. **Separate Variables:** Arrange the equation to isolate \(y\) terms on one side and \(x\) terms on the other.
2. **Integrate Both Sides:** Perform integration on both sides to solve for \(y\).
3. **Solve for \(y^2\):** Introduce the constant \(C\) after integration where necessary and express the solution in terms of \(y^2\).
This method will yield the solution to the differential equation, providing insights into how the functions \(y\) and \(x\) relate based on the equation above.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F30a42a85-c58f-45ac-a4af-faeed1a599e1%2F3f325e18-e471-4cc8-bbd2-c5076c81304e%2Fhp33lp4j_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Solving a Differential Equation Using Separation of Variables**
**Task:**
Using the method of separation of variables, solve the differential equation provided below:
\[
(5 + x^4) \frac{dy}{dx} = \frac{x^3}{y}
\]
Make sure to use **C** to represent the arbitrary constant.
**Solution Format:**
\[ y^2 = \]
[Insert the solution derived from the separation of variables method here.]
**Explanation of Approach:**
To tackle this problem, follow these steps:
1. **Separate Variables:** Arrange the equation to isolate \(y\) terms on one side and \(x\) terms on the other.
2. **Integrate Both Sides:** Perform integration on both sides to solve for \(y\).
3. **Solve for \(y^2\):** Introduce the constant \(C\) after integration where necessary and express the solution in terms of \(y^2\).
This method will yield the solution to the differential equation, providing insights into how the functions \(y\) and \(x\) relate based on the equation above.
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