Solve the following ODE t y" – (1+t)y' + y = t² e' where one complementary solution is y1 = 1+t. %3D
Solve the following ODE t y" – (1+t)y' + y = t² e' where one complementary solution is y1 = 1+t. %3D
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
I got y2 =et, W=tet, W1=-et, and W2=1+t
For my final answer I got c1(1+t)+c2et+et(1-t2/2 + t3/3).
I'm not sure if it is right, but I can't find what, if anything I did wrong.
![### Differential Equations: Solving ODEs
**Problem Statement:**
Solve the following ordinary differential equation (ODE):
\[
t y'' - (1 + t)y' + y = t^2 e^t
\]
where one complementary solution is \( y_1 = 1 + t \).
**Hint:** You may need to use Abel's theorem.
**Detailed Explanation:**
In this problem, you are given a second-order linear differential equation:
\[
t y'' - (1 + t)y' + y = t^2 e^t
\]
and one complementary (particular) solution \( y_1 = 1 + t \).
### Steps to Solve:
1. **Substitute the given complementary solution** into the homogeneous part of the differential equation to verify it.
2. **Apply Abel's Theorem** if necessary to find the second linearly independent solution.
3. **Find the particular integral** of the differential equation using methods such as undetermined coefficients or variation of parameters.
4. **Combine the complementary solution and the particular integral** to form the general solution.
This problem provides a hint that Abel's theorem might be helpful. Abel's theorem is useful for solving linear differential equations and finding the Wronskian of the solutions.
For a detailed step-by-step solution, please refer to the relevant sections of your differential equations textbook or consult with your instructor.
**Keywords:** Ordinary Differential Equation, Complementary Solution, Abel's Theorem, Wronskian, Particular Integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F88e1e2e4-888b-4182-8c02-fd46dda7f6b1%2F729832bb-a9d6-4950-85f8-b604ae2feed2%2Fd0mqotq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Differential Equations: Solving ODEs
**Problem Statement:**
Solve the following ordinary differential equation (ODE):
\[
t y'' - (1 + t)y' + y = t^2 e^t
\]
where one complementary solution is \( y_1 = 1 + t \).
**Hint:** You may need to use Abel's theorem.
**Detailed Explanation:**
In this problem, you are given a second-order linear differential equation:
\[
t y'' - (1 + t)y' + y = t^2 e^t
\]
and one complementary (particular) solution \( y_1 = 1 + t \).
### Steps to Solve:
1. **Substitute the given complementary solution** into the homogeneous part of the differential equation to verify it.
2. **Apply Abel's Theorem** if necessary to find the second linearly independent solution.
3. **Find the particular integral** of the differential equation using methods such as undetermined coefficients or variation of parameters.
4. **Combine the complementary solution and the particular integral** to form the general solution.
This problem provides a hint that Abel's theorem might be helpful. Abel's theorem is useful for solving linear differential equations and finding the Wronskian of the solutions.
For a detailed step-by-step solution, please refer to the relevant sections of your differential equations textbook or consult with your instructor.
**Keywords:** Ordinary Differential Equation, Complementary Solution, Abel's Theorem, Wronskian, Particular Integral.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
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 4 steps with 4 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
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](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
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…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
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…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)