Use the integrating factor method to solve 2xy + y = 2x5/2
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
![**Using the Integrating Factor Method**
We are tasked with solving the differential equation below using the integrating factor method:
\[ 2xy' + y = 2x^{5/2} \]
**Steps to Solve:**
1. **Identify the Standard Form:**
The equation should be in the form \( y' + P(x)y = Q(x) \).
2. **Convert Equation:**
Divide through by \( 2x \) to make it:
\[ y' + \frac{1}{2x}y = x^{3/2} \]
3. **Find the Integrating Factor:**
\[\mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{1}{2x} \, dx} = e^{(\frac{1}{2} \ln x)} \]
Simplify:
\[ \mu(x) = x^{1/2} \]
4. **Multiply Through by Integrating Factor:**
\[ x^{1/2} y' + \frac{x^{1/2}}{2x} y = x^{2} \]
5. **Observe as Exact:**
The left side becomes the derivative of \( (x^{1/2}y) \):
\[ \frac{d}{dx}(x^{1/2} y) = x^2 \]
6. **Integrate Both Sides:**
Integrate with respect to \( x \):
\[ x^{1/2} y = \int x^2 \, dx = \frac{x^3}{3} + C \]
7. **Solve for \( y \):**
\[ y = \frac{x^{3/2}}{3} + Cx^{-1/2} \]
This gives the general solution to the differential equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1d2ea031-a982-4626-b386-36789c43856b%2F8db1dd29-ea9d-4d79-b6c2-b503cde1beac%2Fw8kwuxr_processed.png&w=3840&q=75)
Transcribed Image Text:**Using the Integrating Factor Method**
We are tasked with solving the differential equation below using the integrating factor method:
\[ 2xy' + y = 2x^{5/2} \]
**Steps to Solve:**
1. **Identify the Standard Form:**
The equation should be in the form \( y' + P(x)y = Q(x) \).
2. **Convert Equation:**
Divide through by \( 2x \) to make it:
\[ y' + \frac{1}{2x}y = x^{3/2} \]
3. **Find the Integrating Factor:**
\[\mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{1}{2x} \, dx} = e^{(\frac{1}{2} \ln x)} \]
Simplify:
\[ \mu(x) = x^{1/2} \]
4. **Multiply Through by Integrating Factor:**
\[ x^{1/2} y' + \frac{x^{1/2}}{2x} y = x^{2} \]
5. **Observe as Exact:**
The left side becomes the derivative of \( (x^{1/2}y) \):
\[ \frac{d}{dx}(x^{1/2} y) = x^2 \]
6. **Integrate Both Sides:**
Integrate with respect to \( x \):
\[ x^{1/2} y = \int x^2 \, dx = \frac{x^3}{3} + C \]
7. **Solve for \( y \):**
\[ y = \frac{x^{3/2}}{3} + Cx^{-1/2} \]
This gives the general solution to the differential equation.
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