Solue (x+1)y" + xy -y =0 ; yco)2, 4'c0)= i ginen that either e* or e-x is a solution of (x+D y" + xy'-Y=0._
Solue (x+1)y" + xy -y =0 ; yco)2, 4'c0)= i ginen that either e* or e-x is a solution of (x+D y" + xy'-Y=0._
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
100%
![### Problem Statement
Given the differential equation:
\[ (x+1)y'' + xy' - y = 0 \]
with the initial conditions:
\[ y(0) = 2 \]
\[ y'(0) = -1 \]
and the information that either \( e^x \) or \( e^{-x} \) is a solution of the differential equation, solve for \( y \).
### Explanation of the Differential Equation
This is a second-order linear differential equation with non-constant coefficients. The equation is:
\[ (x+1)y'' + xy' - y = 0 \]
where:
- \( y'' \) is the second derivative of \( y \) with respect to \( x \),
- \( y' \) is the first derivative of \( y \) with respect to \( x \),
- \( y \) is the function to be determined.
### Initial Conditions
- \( y(0) = 2 \): The value of the function \( y \) when \( x = 0 \) is 2.
- \( y'(0) = -1 \): The value of the derivative of \( y \) when \( x = 0 \) is -1.
### Given Solutions
It is given that either \( e^x \) or \( e^{-x} \) is a solution to the differential equation. This implies that the solution for \( y \) could involve these exponential functions.
### Approach
1. **Identify the Homogeneous and Particular Solutions**:
- Given the nature of the differential equation and the provided potential solutions, substitute \( e^x \) and \( e^{-x} \) into the equation to verify which one satisfies the equation.
2. **Form the General Solution**:
- Construct the general solution based on the verified solutions.
3. **Apply Initial Conditions**:
- Use the initial conditions to determine the specific coefficients in the general solution.
### Resources
For a detailed step-by-step solution, consult your differential equations textbook or online resources on solving second-order linear differential equations with variable coefficients.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F72bb43eb-cb6a-499b-b213-8a074a437ef9%2F840b0451-a280-4f91-8da7-9666bd5dd81a%2Fedysfk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Given the differential equation:
\[ (x+1)y'' + xy' - y = 0 \]
with the initial conditions:
\[ y(0) = 2 \]
\[ y'(0) = -1 \]
and the information that either \( e^x \) or \( e^{-x} \) is a solution of the differential equation, solve for \( y \).
### Explanation of the Differential Equation
This is a second-order linear differential equation with non-constant coefficients. The equation is:
\[ (x+1)y'' + xy' - y = 0 \]
where:
- \( y'' \) is the second derivative of \( y \) with respect to \( x \),
- \( y' \) is the first derivative of \( y \) with respect to \( x \),
- \( y \) is the function to be determined.
### Initial Conditions
- \( y(0) = 2 \): The value of the function \( y \) when \( x = 0 \) is 2.
- \( y'(0) = -1 \): The value of the derivative of \( y \) when \( x = 0 \) is -1.
### Given Solutions
It is given that either \( e^x \) or \( e^{-x} \) is a solution to the differential equation. This implies that the solution for \( y \) could involve these exponential functions.
### Approach
1. **Identify the Homogeneous and Particular Solutions**:
- Given the nature of the differential equation and the provided potential solutions, substitute \( e^x \) and \( e^{-x} \) into the equation to verify which one satisfies the equation.
2. **Form the General Solution**:
- Construct the general solution based on the verified solutions.
3. **Apply Initial Conditions**:
- Use the initial conditions to determine the specific coefficients in the general solution.
### Resources
For a detailed step-by-step solution, consult your differential equations textbook or online resources on solving second-order linear differential equations with variable coefficients.
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 2 steps with 4 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.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,
