У - 2у+ %3D 1+ x
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
Solve the given Differential Equation by variation of parameters.
![The equation shown is a second-order linear differential equation:
\[ y'' - 2y' + y = \frac{e^x}{1 + x^2} \]
### Explanation:
- **\( y'' \)**: This term represents the second derivative of \( y \) with respect to \( x \), indicating how the rate of change of the rate of change of \( y \) is affected.
- **\(- 2y'\)**: This is the first derivative of \( y \) multiplied by -2, which implies a damping or reducing effect on the system described by this equation.
- **\( + y \)**: This term keeps the function \( y \) itself in the equation, acting as feedback in the system.
- **\[ = \frac{e^x}{1 + x^2} \]**: The right-hand side of the equation is a function of \( x \). Here, \( e^x \) is the exponential function, while \( 1 + x^2 \) is the denominator, suggesting the equation has a response to an exponential input that is moderated by a polynomial factor.
This equation may be used in contexts such as physics or engineering to model complex systems where change occurs over time. Solving it typically involves finding a particular solution to the inhomogeneous equation alongside the general solution of the corresponding homogeneous equation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F35ecb4dc-8e06-4359-aba9-116b0df055b7%2F88d8914a-679d-495a-b8d9-7a627db1a652%2Fwqvofix_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The equation shown is a second-order linear differential equation:
\[ y'' - 2y' + y = \frac{e^x}{1 + x^2} \]
### Explanation:
- **\( y'' \)**: This term represents the second derivative of \( y \) with respect to \( x \), indicating how the rate of change of the rate of change of \( y \) is affected.
- **\(- 2y'\)**: This is the first derivative of \( y \) multiplied by -2, which implies a damping or reducing effect on the system described by this equation.
- **\( + y \)**: This term keeps the function \( y \) itself in the equation, acting as feedback in the system.
- **\[ = \frac{e^x}{1 + x^2} \]**: The right-hand side of the equation is a function of \( x \). Here, \( e^x \) is the exponential function, while \( 1 + x^2 \) is the denominator, suggesting the equation has a response to an exponential input that is moderated by a polynomial factor.
This equation may be used in contexts such as physics or engineering to model complex systems where change occurs over time. Solving it typically involves finding a particular solution to the inhomogeneous equation alongside the general solution of the corresponding homogeneous equation.
Expert Solution
Step 1
Step by step
Solved in 2 steps with 2 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,
