Solve the following differential equation (you may leave your answer in y+x²y=x²
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![---
**Differential Equations Course**
**Chapter: Solving Differential Equations**
### Example Problem
**Solve the following differential equation (you may leave your answer in implicit form):**
\[ y' + x^2 y = x^2 \]
---
In this example, we are dealing with a first-order linear differential equation. To solve this, we can use the method of integrating factors. The standard form of a first-order linear differential equation is:
\[ y' + P(x)y = Q(x) \]
Here, \( P(x) = x^2 \) and \( Q(x) = x^2 \).
To solve it, follow these steps:
1. Find the integrating factor, \(\mu(x)\), which is given by:
\[
\mu(x) = e^{\int P(x) \, dx} = e^{\int x^2 \, dx} = e^{\frac{x^3}{3}}
\]
2. Multiply both sides of the differential equation by the integrating factor:
\[
e^{\frac{x^3}{3}}y' + e^{\frac{x^3}{3}}x^2 y = e^{\frac{x^3}{3}} x^2
\]
3. The left side of the equation becomes the derivative of \( y \cdot \mu(x) \):
\[
\frac{d}{dx}\left( e^{\frac{x^3}{3}} y \right) = e^{\frac{x^3}{3}} x^2
\]
4. Integrate both sides with respect to \( x \):
\[
e^{\frac{x^3}{3}} y = \int e^{\frac{x^3}{3}} x^2 \, dx
\]
5. Use substitution to solve the integral on the right side. Let \( u = \frac{x^3}{3} \), thus \( du = x^2 dx \), and the integral becomes:
\[
\int e^u \, du = e^u + C
\]
Substituting back \( u = \frac{x^3}{3} \):
\[
e^{\frac{x^3}{3}} y = e^{\frac{x^3}{3}} + C
\]
6.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe57d9078-3a26-4ab9-a5c6-f67f173c178d%2F6be1fd6b-fcb4-4776-ae22-175bf8fe6926%2Fg8mvdff_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
**Differential Equations Course**
**Chapter: Solving Differential Equations**
### Example Problem
**Solve the following differential equation (you may leave your answer in implicit form):**
\[ y' + x^2 y = x^2 \]
---
In this example, we are dealing with a first-order linear differential equation. To solve this, we can use the method of integrating factors. The standard form of a first-order linear differential equation is:
\[ y' + P(x)y = Q(x) \]
Here, \( P(x) = x^2 \) and \( Q(x) = x^2 \).
To solve it, follow these steps:
1. Find the integrating factor, \(\mu(x)\), which is given by:
\[
\mu(x) = e^{\int P(x) \, dx} = e^{\int x^2 \, dx} = e^{\frac{x^3}{3}}
\]
2. Multiply both sides of the differential equation by the integrating factor:
\[
e^{\frac{x^3}{3}}y' + e^{\frac{x^3}{3}}x^2 y = e^{\frac{x^3}{3}} x^2
\]
3. The left side of the equation becomes the derivative of \( y \cdot \mu(x) \):
\[
\frac{d}{dx}\left( e^{\frac{x^3}{3}} y \right) = e^{\frac{x^3}{3}} x^2
\]
4. Integrate both sides with respect to \( x \):
\[
e^{\frac{x^3}{3}} y = \int e^{\frac{x^3}{3}} x^2 \, dx
\]
5. Use substitution to solve the integral on the right side. Let \( u = \frac{x^3}{3} \), thus \( du = x^2 dx \), and the integral becomes:
\[
\int e^u \, du = e^u + C
\]
Substituting back \( u = \frac{x^3}{3} \):
\[
e^{\frac{x^3}{3}} y = e^{\frac{x^3}{3}} + C
\]
6.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
