Obtain the general solution to the equation. (x²+6) + xy - 8x = 0 dy dx The general solution is y(x) = ignoring lost solutions, if any.
Obtain the general solution to the equation. (x²+6) + xy - 8x = 0 dy dx The general solution is y(x) = ignoring lost solutions, if any.
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 Statement:**
**Obtain the general solution to the equation:**
\[ \left( x^2 + 6 \right) \frac{dy}{dx} + xy - 8x = 0 \]
---
**Solution Procedure:**
**Step 1: Rewrite the equation**
Rewrite the given equation in standard form:
\[ \left( x^2 + 6 \right) \frac{dy}{dx} + xy = 8x \]
**Step 2: Identify the type of differential equation**
This is a linear first-order differential equation of the form:
\[ P(x) \frac{dy}{dx} + Q(x)y = R(x) \]
Where:
- \( P(x) = x^2 + 6 \)
- \( Q(x) = x \)
- \( R(x) = 8x \)
**Step 3: Find the integrating factor**
To solve this differential equation, find the integrating factor \( \mu(x) \):
\[ \mu(x) = e^{\int \frac{Q(x)}{P(x)} dx} = e^{\int \frac{x}{x^2 + 6} dx} \]
Using substitution \( u = x^2 + 6 \), then \( du = 2x dx \), the integral simplifies to:
\[ \mu(x) = e^{\frac{1}{2} \ln|x^2 + 6|} = (x^2 + 6)^{\frac{1}{2}} = \sqrt{x^2 + 6} \]
**Step 4: Multiply through by the integrating factor**
Multiply both sides of the differential equation by the integrating factor:
\[ \sqrt{x^2 + 6}(x^2 + 6) \frac{dy}{dx} + \sqrt{x^2 + 6} \cdot xy = 8x \sqrt{x^2 + 6} \]
**Step 5: Simplify and integrate**
The left-hand side of the equation is the derivative of \( y \mu(x) \):
\[ \frac{d}{dx} \left( y \sqrt{x^2 + 6} \right) = 8x \sqrt{x^2 + 6} \]
Integrate both sides with respect to \( x \):
\[ y \sqrt{x^2 + 6} = \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1887966a-d451-4216-a13a-dc9720678597%2Fea5f7b63-0b75-4189-a0cd-398176cbad18%2F05b7c99_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
**Obtain the general solution to the equation:**
\[ \left( x^2 + 6 \right) \frac{dy}{dx} + xy - 8x = 0 \]
---
**Solution Procedure:**
**Step 1: Rewrite the equation**
Rewrite the given equation in standard form:
\[ \left( x^2 + 6 \right) \frac{dy}{dx} + xy = 8x \]
**Step 2: Identify the type of differential equation**
This is a linear first-order differential equation of the form:
\[ P(x) \frac{dy}{dx} + Q(x)y = R(x) \]
Where:
- \( P(x) = x^2 + 6 \)
- \( Q(x) = x \)
- \( R(x) = 8x \)
**Step 3: Find the integrating factor**
To solve this differential equation, find the integrating factor \( \mu(x) \):
\[ \mu(x) = e^{\int \frac{Q(x)}{P(x)} dx} = e^{\int \frac{x}{x^2 + 6} dx} \]
Using substitution \( u = x^2 + 6 \), then \( du = 2x dx \), the integral simplifies to:
\[ \mu(x) = e^{\frac{1}{2} \ln|x^2 + 6|} = (x^2 + 6)^{\frac{1}{2}} = \sqrt{x^2 + 6} \]
**Step 4: Multiply through by the integrating factor**
Multiply both sides of the differential equation by the integrating factor:
\[ \sqrt{x^2 + 6}(x^2 + 6) \frac{dy}{dx} + \sqrt{x^2 + 6} \cdot xy = 8x \sqrt{x^2 + 6} \]
**Step 5: Simplify and integrate**
The left-hand side of the equation is the derivative of \( y \mu(x) \):
\[ \frac{d}{dx} \left( y \sqrt{x^2 + 6} \right) = 8x \sqrt{x^2 + 6} \]
Integrate both sides with respect to \( x \):
\[ y \sqrt{x^2 + 6} = \
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