Solve the given differential equation by undetermined coefficients. y" - 12y' + 36y = 12x + 6 y(x)
Solve the given differential equation by undetermined coefficients. y" - 12y' + 36y = 12x + 6 y(x)
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 by Undetermined Coefficients
### Problem Statement
Solve the given differential equation by the method of undetermined coefficients:
\[ y'' - 12y' + 36y = 12x + 6 \]
\[ y(x) = \_\_\_\_\_\_\_\_\_ \]
### Explanation
In this problem, we are tasked with solving a non-homogeneous second-order linear differential equation using the method of undetermined coefficients. This method is particularly useful for solving linear differential equations with constant coefficients and a specific type of non-homogeneous term.
Steps involved in solving the differential equation:
1. **Solve the Homogeneous Equation**:
First, find the general solution to the associated homogeneous differential equation:
\[ y'' - 12y' + 36y = 0 \]
2. **Find the Particular Solution**:
Next, we find a particular solution to the original non-homogeneous differential equation. The form of the particular solution depends on the non-homogeneous term \(12x + 6\).
3. **Combine Both Solutions**:
The general solution of the original non-homogeneous differential equation is the sum of the general solution of the homogeneous equation and the particular solution.
### Detailed Steps
1. **Homogeneous Solution**:
The characteristic equation for \( y'' - 12y' + 36y = 0 \) is:
\[ r^2 - 12r + 36 = 0 \]
Solving for \(r\), we get:
\[ (r - 6)^2 = 0 \]
So, \( r = 6 \) (a repeated root).
Therefore, the general solution to the homogeneous equation is:
\[ y_h(x) = (C_1 + C_2x)e^{6x} \]
2. **Particular Solution**:
Assume a particular solution of the form \( y_p(x) = Ax + B \) since the non-homogeneous term is \(12x + 6\).
Differentiate \( y_p \):
\[ y_p'(x) = A \]
\[ y_p''(x) = 0 \]
Substitute \( y_p(x) \), \( y_p'(x) \), and \( y_p''(x) \) into the original](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F995b5b40-26b1-420f-ae4c-d7493773d7f9%2F164af3a7-8522-4d56-b509-60ccf078b084%2F7pb1g0b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Solving Differential Equations by Undetermined Coefficients
### Problem Statement
Solve the given differential equation by the method of undetermined coefficients:
\[ y'' - 12y' + 36y = 12x + 6 \]
\[ y(x) = \_\_\_\_\_\_\_\_\_ \]
### Explanation
In this problem, we are tasked with solving a non-homogeneous second-order linear differential equation using the method of undetermined coefficients. This method is particularly useful for solving linear differential equations with constant coefficients and a specific type of non-homogeneous term.
Steps involved in solving the differential equation:
1. **Solve the Homogeneous Equation**:
First, find the general solution to the associated homogeneous differential equation:
\[ y'' - 12y' + 36y = 0 \]
2. **Find the Particular Solution**:
Next, we find a particular solution to the original non-homogeneous differential equation. The form of the particular solution depends on the non-homogeneous term \(12x + 6\).
3. **Combine Both Solutions**:
The general solution of the original non-homogeneous differential equation is the sum of the general solution of the homogeneous equation and the particular solution.
### Detailed Steps
1. **Homogeneous Solution**:
The characteristic equation for \( y'' - 12y' + 36y = 0 \) is:
\[ r^2 - 12r + 36 = 0 \]
Solving for \(r\), we get:
\[ (r - 6)^2 = 0 \]
So, \( r = 6 \) (a repeated root).
Therefore, the general solution to the homogeneous equation is:
\[ y_h(x) = (C_1 + C_2x)e^{6x} \]
2. **Particular Solution**:
Assume a particular solution of the form \( y_p(x) = Ax + B \) since the non-homogeneous term is \(12x + 6\).
Differentiate \( y_p \):
\[ y_p'(x) = A \]
\[ y_p''(x) = 0 \]
Substitute \( y_p(x) \), \( y_p'(x) \), and \( y_p''(x) \) into the original
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