1. Consider the equation -xy"+y' + (x-1)y=0, x > 0. We can easily verify that y₁(x) = e* is a solution of the equation. Use reduction of order to determine the general solution of the equation.
1. Consider the equation -xy"+y' + (x-1)y=0, x > 0. We can easily verify that y₁(x) = e* is a solution of the equation. Use reduction of order to determine the general solution of the equation.
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: Reduction of Order
#### Example Problem:
1. Consider the equation \( -xy'' + y' + (x - 1)y = 0 \), where \( x > 0 \). We can easily verify that \( y_1(x) = e^x \) is a solution of the equation. Use reduction of order to determine the general solution of the equation.
To solve this differential equation, we will use the method of reduction of order. This technique is useful when one solution to a second-order linear differential equation is already known. In this case, we know that \( y_1(x) = e^x \) is a solution.
Let's assume a second solution of the form
\[ y_2(x) = v(x)e^x \]
where \( v(x) \) is a function to be determined. By substituting \( y_2(x) \) into the original differential equation and simplifying, we can find \( v(x) \) and thus the general solution.
### Steps to Solve Using Reduction of Order:
1. **Substitute** \( y_2(x) = v(x)e^x \) into the differential equation.
2. **Differentiate** to find expressions for \( y'_2 \) and \( y''_2 \).
3. **Simplify** the resulting equation, using the fact that \( e^x \) is a known solution.
4. **Solve** for \( v(x) \).
5. **Integrate** to find the general solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24e9c030-c4e6-49da-b6c6-4b7d6c7f6fcd%2F4424f660-1e6b-45ba-867f-3b1e5b52a0f6%2Fiai9rrp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Differential Equations: Reduction of Order
#### Example Problem:
1. Consider the equation \( -xy'' + y' + (x - 1)y = 0 \), where \( x > 0 \). We can easily verify that \( y_1(x) = e^x \) is a solution of the equation. Use reduction of order to determine the general solution of the equation.
To solve this differential equation, we will use the method of reduction of order. This technique is useful when one solution to a second-order linear differential equation is already known. In this case, we know that \( y_1(x) = e^x \) is a solution.
Let's assume a second solution of the form
\[ y_2(x) = v(x)e^x \]
where \( v(x) \) is a function to be determined. By substituting \( y_2(x) \) into the original differential equation and simplifying, we can find \( v(x) \) and thus the general solution.
### Steps to Solve Using Reduction of Order:
1. **Substitute** \( y_2(x) = v(x)e^x \) into the differential equation.
2. **Differentiate** to find expressions for \( y'_2 \) and \( y''_2 \).
3. **Simplify** the resulting equation, using the fact that \( e^x \) is a known solution.
4. **Solve** for \( v(x) \).
5. **Integrate** to find the general solution.
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 3 steps with 8 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,
