xy’ + (1+2x?)y = x3e-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: Separable or Linear
#### Instructions:
Solve the following equations. Each question is either separable or linear.
1. [Redacted]
2. [Redacted]
3. [Redacted]
4. \( xy' + (1 + 2x^2)y = x^3e^{-x^2} \)
#### Explanation of Terms:
1. **Separable Equations**: Differential equations in which variables can be separated on either side of the equation. For example, any equation that can be rearranged into the form \(g(y)dy = f(x)dx\).
2. **Linear Equations**: First-order differential equations of the form \( y' + p(x)y = q(x) \). These equations can often be solved using an integrating factor.
#### Example Problem:
Consider the provided example:
\[ xy' + (1 + 2x^2)y = x^3e^{-x^2} \]
To solve this, follow these steps:
1. **Identify the Standard Form**:
The standard form of a linear differential equation is \( y' + p(x)y = q(x) \). Rewrite the given equation in this form:
\[
y' + \left(\frac{1 + 2x^2}{x}\right)y = x^2e^{-x^2}
\]
2. **Find Integrating Factor**:
Compute the integrating factor \( \mu(x) \):
\[
\mu(x) = e^{\int \frac{1 + 2x^2}{x} dx} = e^{\ln|x| + \frac{2x^2}{2}} = x \cdot e^{x^2}
\]
3. **Multiply Through by Integrating Factor**:
Multiply the entire differential equation by the integrating factor:
\[
x e^{x^2} \cdot y' + (1 + 2x^2) e^{x^2} y = x^2 e^{-x^2} \cdot x e^{x^2}
\]
Simplifies to:
\[
\frac{d}{dx} [y \cdot xe^{x^2}] = x^3
\]
4. **Integrate Both Sides**:
Integrate both sides with respect to x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac0f3c56-5561-4b47-8ddf-f02546f51c3e%2Fb345568d-e951-428a-96be-f13734ed03f6%2Fca3tiq_processed.png&w=3840&q=75)
Transcribed Image Text:### Differential Equations: Separable or Linear
#### Instructions:
Solve the following equations. Each question is either separable or linear.
1. [Redacted]
2. [Redacted]
3. [Redacted]
4. \( xy' + (1 + 2x^2)y = x^3e^{-x^2} \)
#### Explanation of Terms:
1. **Separable Equations**: Differential equations in which variables can be separated on either side of the equation. For example, any equation that can be rearranged into the form \(g(y)dy = f(x)dx\).
2. **Linear Equations**: First-order differential equations of the form \( y' + p(x)y = q(x) \). These equations can often be solved using an integrating factor.
#### Example Problem:
Consider the provided example:
\[ xy' + (1 + 2x^2)y = x^3e^{-x^2} \]
To solve this, follow these steps:
1. **Identify the Standard Form**:
The standard form of a linear differential equation is \( y' + p(x)y = q(x) \). Rewrite the given equation in this form:
\[
y' + \left(\frac{1 + 2x^2}{x}\right)y = x^2e^{-x^2}
\]
2. **Find Integrating Factor**:
Compute the integrating factor \( \mu(x) \):
\[
\mu(x) = e^{\int \frac{1 + 2x^2}{x} dx} = e^{\ln|x| + \frac{2x^2}{2}} = x \cdot e^{x^2}
\]
3. **Multiply Through by Integrating Factor**:
Multiply the entire differential equation by the integrating factor:
\[
x e^{x^2} \cdot y' + (1 + 2x^2) e^{x^2} y = x^2 e^{-x^2} \cdot x e^{x^2}
\]
Simplifies to:
\[
\frac{d}{dx} [y \cdot xe^{x^2}] = x^3
\]
4. **Integrate Both Sides**:
Integrate both sides with respect to x
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
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,
