Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem 3: Solving an ODE Using the Method of Undetermined Coefficients**
In this problem, we will employ the method of undetermined coefficients, as explained in section 3.5, to determine the general solution of the given ordinary differential equation (ODE):
\[ y'' - 2y' + y = e^t + 1. \]
### Explanation
- **Method of Undetermined Coefficients:** This method is used to find particular solutions to linear constant coefficient differential equations. The idea is to guess a form for the particular solution based on the non-homogeneous term and then determine the coefficients by substituting back into the differential equation.
- **Equation Components:**
- \( y'' \): the second derivative of \( y \) with respect to \( t \).
- \( y' \): the first derivative of \( y \) with respect to \( t \).
- \( y \): the function of \( t \) we aim to find.
- \( e^t + 1 \): the non-homogeneous part, which will guide us in guessing the particular solution.
### Steps to Solve:
1. **Characteristic Equation:** Determine the characteristic equation for the homogeneous part of the ODE: \( y'' - 2y' + y = 0 \).
2. **Homogeneous Solution:** Solve the characteristic equation to find the complementary or homogeneous solution, \( y_h \).
3. **Particular Solution:** Guess a form for the particular solution, \( y_p \), based on the non-homogeneous term \( e^t + 1 \).
4. **Solution Verification:** Substitute \( y_h + y_p \) back into the original ODE to ensure it satisfies it.
5. **General Solution:** Combine \( y_h \) and \( y_p \) to form the general solution of the ODE.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5b55a0fb-5dff-4587-97c0-c3ec381e9c7e%2Fda0ce568-9c3f-4350-b1c1-f291c0dfdffc%2Fbh007ys_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3: Solving an ODE Using the Method of Undetermined Coefficients**
In this problem, we will employ the method of undetermined coefficients, as explained in section 3.5, to determine the general solution of the given ordinary differential equation (ODE):
\[ y'' - 2y' + y = e^t + 1. \]
### Explanation
- **Method of Undetermined Coefficients:** This method is used to find particular solutions to linear constant coefficient differential equations. The idea is to guess a form for the particular solution based on the non-homogeneous term and then determine the coefficients by substituting back into the differential equation.
- **Equation Components:**
- \( y'' \): the second derivative of \( y \) with respect to \( t \).
- \( y' \): the first derivative of \( y \) with respect to \( t \).
- \( y \): the function of \( t \) we aim to find.
- \( e^t + 1 \): the non-homogeneous part, which will guide us in guessing the particular solution.
### Steps to Solve:
1. **Characteristic Equation:** Determine the characteristic equation for the homogeneous part of the ODE: \( y'' - 2y' + y = 0 \).
2. **Homogeneous Solution:** Solve the characteristic equation to find the complementary or homogeneous solution, \( y_h \).
3. **Particular Solution:** Guess a form for the particular solution, \( y_p \), based on the non-homogeneous term \( e^t + 1 \).
4. **Solution Verification:** Substitute \( y_h + y_p \) back into the original ODE to ensure it satisfies it.
5. **General Solution:** Combine \( y_h \) and \( y_p \) to form the general solution of the ODE.
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