Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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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.
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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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