By using variation of parameters, determine the general solution of t²y" - 3ty' +4y=t, t > 0 given that y₁ (t) = t², 3₂(t) = t² lnt constitute a fundamental set of solutions of the homogeneous equation.
By using variation of parameters, determine the general solution of t²y" - 3ty' +4y=t, t > 0 given that y₁ (t) = t², 3₂(t) = t² lnt constitute a fundamental set of solutions of the homogeneous 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
![**Title: Variation of Parameters: General Solution to Differential Equation**
**Problem Statement:**
Determine the general solution of the differential equation
\[ t^2 y'' - 3t y' + 4y = t^5, \;\; t > 0 \]
by using the method of variation of parameters, given that
\[ y_1(t) = t^2 \quad \text{and} \quad y_2(t) = t^2 \ln t \]
constitute a fundamental set of solutions of the homogeneous equation.
---
**Explanation:**
To solve this differential equation using the variation of parameters, follow these steps:
1. **Identify Homogeneous Solutions:**
The given solutions \( y_1(t) \) and \( y_2(t) \) are solutions to the corresponding homogeneous equation:
\[ t^2 y'' - 3t y' + 4y = 0 \]
2. **Setup the General Solution:**
Assume the general solution of the non-homogeneous equation is of the form:
\[ y(t) = u_1(t) y_1(t) + u_2(t) y_2(t) \]
where \( u_1(t) \) and \( u_2(t) \) are functions to be determined.
3. **Calculate Derivatives:**
Compute the first and second derivatives:
\[ y' = u_1' y_1 + u_1 y_1' + u_2' y_2 + u_2 y_2' \]
\[ y'' = u_1'' y_1 + 2u_1' y_1' + u_1 y_1'' + u_2'' y_2 + 2u_2' y_2' + u_2 y_2'' \]
4. **Plug Into Original Equation:**
Substitute these into the original differential equation and equate coefficients.
5. **Determine \( u_1(t) \) and \( u_2(t) \):**
Solve the resulting system of equations to find expressions for \( u_1 \) and \( u_2 \).
6. **Integrate:**
Integrate to find the explicit forms of \( u_1(t) \) and \( u_2(t) \).
7. **Construct the Final Solution](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc47a605b-1167-4b41-bc40-883b330bf596%2Fbde63670-611a-4f38-ae34-66686553d6b6%2Fti6f9bk_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Variation of Parameters: General Solution to Differential Equation**
**Problem Statement:**
Determine the general solution of the differential equation
\[ t^2 y'' - 3t y' + 4y = t^5, \;\; t > 0 \]
by using the method of variation of parameters, given that
\[ y_1(t) = t^2 \quad \text{and} \quad y_2(t) = t^2 \ln t \]
constitute a fundamental set of solutions of the homogeneous equation.
---
**Explanation:**
To solve this differential equation using the variation of parameters, follow these steps:
1. **Identify Homogeneous Solutions:**
The given solutions \( y_1(t) \) and \( y_2(t) \) are solutions to the corresponding homogeneous equation:
\[ t^2 y'' - 3t y' + 4y = 0 \]
2. **Setup the General Solution:**
Assume the general solution of the non-homogeneous equation is of the form:
\[ y(t) = u_1(t) y_1(t) + u_2(t) y_2(t) \]
where \( u_1(t) \) and \( u_2(t) \) are functions to be determined.
3. **Calculate Derivatives:**
Compute the first and second derivatives:
\[ y' = u_1' y_1 + u_1 y_1' + u_2' y_2 + u_2 y_2' \]
\[ y'' = u_1'' y_1 + 2u_1' y_1' + u_1 y_1'' + u_2'' y_2 + 2u_2' y_2' + u_2 y_2'' \]
4. **Plug Into Original Equation:**
Substitute these into the original differential equation and equate coefficients.
5. **Determine \( u_1(t) \) and \( u_2(t) \):**
Solve the resulting system of equations to find expressions for \( u_1 \) and \( u_2 \).
6. **Integrate:**
Integrate to find the explicit forms of \( u_1(t) \) and \( u_2(t) \).
7. **Construct the Final 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 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,
