Solve the given D.E. by the method of undetermined coefficients. (Find y = ye+ yp) y" - 7y +12y = 5e²
Solve the given D.E. by the method of undetermined coefficients. (Find y = ye+ yp) y" - 7y +12y = 5e²
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Solving Differential Equations Using the Method of Undetermined Coefficients
**Objective:** Solve the given differential equation (D.E.) by the method of undetermined coefficients.
**Task:** Find \( y \) where \( y = y_c + y_p \)
The differential equation to solve is:
\[ y'' - 7y' + 12y = 5e^{2x} \]
#### Explanation:
1. **\( y_c \)** is the complementary solution, which solves the homogeneous equation \( y'' - 7y' + 12y = 0 \).
2. **\( y_p \)** is the particular solution, which solves the non-homogeneous equation given above \( y'' - 7y' + 12y = 5e^{2x} \).
### Steps to Solve:
#### 1. Finding the Complementary Solution \( y_c \):
- Solve the characteristic equation for the homogeneous part:
\[ r^2 - 7r + 12 = 0 \]
- Determine the roots of the characteristic equation.
#### 2. Finding the Particular Solution \( y_p \):
- Since the non-homogeneous term is \( 5e^{2x} \), guess a particular solution of the form \( y_p = Ae^{2x} \).
- Substitute \( y_p \) into the non-homogeneous differential equation to solve for \( A \).
#### 3. Combine the results to obtain the general solution:
\[ y = y_c + y_p \]
This method is particularly useful for linear differential equations with constant coefficients where the non-homogeneous term is of a specific type (e.g., exponential functions, polynomials, or sine/cosine functions).
Keep practicing different types of non-homogeneous terms to improve your skills in solving differential equations using the method of undetermined coefficients.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4b62327-b6c2-4927-99ce-af68105c93e2%2F4ba84a2e-b82c-4cb1-843c-6e3319aa69de%2F61eiwh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving Differential Equations Using the Method of Undetermined Coefficients
**Objective:** Solve the given differential equation (D.E.) by the method of undetermined coefficients.
**Task:** Find \( y \) where \( y = y_c + y_p \)
The differential equation to solve is:
\[ y'' - 7y' + 12y = 5e^{2x} \]
#### Explanation:
1. **\( y_c \)** is the complementary solution, which solves the homogeneous equation \( y'' - 7y' + 12y = 0 \).
2. **\( y_p \)** is the particular solution, which solves the non-homogeneous equation given above \( y'' - 7y' + 12y = 5e^{2x} \).
### Steps to Solve:
#### 1. Finding the Complementary Solution \( y_c \):
- Solve the characteristic equation for the homogeneous part:
\[ r^2 - 7r + 12 = 0 \]
- Determine the roots of the characteristic equation.
#### 2. Finding the Particular Solution \( y_p \):
- Since the non-homogeneous term is \( 5e^{2x} \), guess a particular solution of the form \( y_p = Ae^{2x} \).
- Substitute \( y_p \) into the non-homogeneous differential equation to solve for \( A \).
#### 3. Combine the results to obtain the general solution:
\[ y = y_c + y_p \]
This method is particularly useful for linear differential equations with constant coefficients where the non-homogeneous term is of a specific type (e.g., exponential functions, polynomials, or sine/cosine functions).
Keep practicing different types of non-homogeneous terms to improve your skills in solving differential equations using the method of undetermined coefficients.
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