Solve the given differential equation by undetermined coefficients. у" — бу' + 9у %3D 21x + 5 =
Solve the given differential equation by undetermined coefficients. у" — бу' + 9у %3D 21x + 5 =
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
![### Solving Differential Equations by Undetermined Coefficients
In this example, we aim to solve the given differential equation using the method of undetermined coefficients.
The differential equation to solve is:
\[ y'' - 6y' + 9y = 21x + 5 \]
The proposed solution \( y(x) \) is:
\[ y(x) = c_1 e^{3x} + c_2 xe^{3x} - \frac{19}{6} x^2 - \frac{17}{9} x \]
Where \( c_1 \) and \( c_2 \) are constants determined by the initial conditions.
#### Explanation of Terms
- **\(e^{3x}\)**: This represents an exponential function with a base \(e\).
- **\(xe^{3x}\)**: This is the exponential function multiplied by \(x\).
- **Constants \(\frac{19}{6}\) and \(\frac{17}{9}\)**: These are coefficients for the polynomial terms \(x^2\) and \(x\), respectively.
#### Understanding the Solution Components
1. **Homogeneous Solution**:
The terms \( c_1 e^{3x} \) and \( c_2 xe^{3x} \) constitute the complementary solution derived from solving the homogeneous equation \( y'' - 6y' + 9y = 0 \).
2. **Particular Solution**:
The polynomial terms \( -\frac{19}{6} x^2 - \frac{17}{9} x \) form the particular solution that specifically satisfies the non-homogeneous part, \( 21x + 5 \).
### Conclusion
By substituting this general solution into the differential equation, we can confirm that it satisfies the given equation. The constants \( c_1 \) and \( c_2 \) will be determined based on initial conditions or additional criteria provided in specific problems.
This elegant blend of exponential functions and polynomials demonstrates the robust capacity of the method of undetermined coefficients to solve a broad range of differential equations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F19331858-c67c-414f-8328-5459040eae8a%2F3155a4c5-466e-4a5e-9d60-9d2ff1676168%2Fh8l76l.png&w=3840&q=75)
Transcribed Image Text:### Solving Differential Equations by Undetermined Coefficients
In this example, we aim to solve the given differential equation using the method of undetermined coefficients.
The differential equation to solve is:
\[ y'' - 6y' + 9y = 21x + 5 \]
The proposed solution \( y(x) \) is:
\[ y(x) = c_1 e^{3x} + c_2 xe^{3x} - \frac{19}{6} x^2 - \frac{17}{9} x \]
Where \( c_1 \) and \( c_2 \) are constants determined by the initial conditions.
#### Explanation of Terms
- **\(e^{3x}\)**: This represents an exponential function with a base \(e\).
- **\(xe^{3x}\)**: This is the exponential function multiplied by \(x\).
- **Constants \(\frac{19}{6}\) and \(\frac{17}{9}\)**: These are coefficients for the polynomial terms \(x^2\) and \(x\), respectively.
#### Understanding the Solution Components
1. **Homogeneous Solution**:
The terms \( c_1 e^{3x} \) and \( c_2 xe^{3x} \) constitute the complementary solution derived from solving the homogeneous equation \( y'' - 6y' + 9y = 0 \).
2. **Particular Solution**:
The polynomial terms \( -\frac{19}{6} x^2 - \frac{17}{9} x \) form the particular solution that specifically satisfies the non-homogeneous part, \( 21x + 5 \).
### Conclusion
By substituting this general solution into the differential equation, we can confirm that it satisfies the given equation. The constants \( c_1 \) and \( c_2 \) will be determined based on initial conditions or additional criteria provided in specific problems.
This elegant blend of exponential functions and polynomials demonstrates the robust capacity of the method of undetermined coefficients to solve a broad range of differential equations.
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 5 steps with 9 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,
