Use the method of undetermined coefficients to solve the given nonhomogeneous system. 1 4 -7 *-( * )* - ()- X' = 3 + 10 9 6 X(t) =
Use the method of undetermined coefficients to solve the given nonhomogeneous system. 1 4 -7 *-( * )* - ()- X' = 3 + 10 9 6 X(t) =
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 a Nonhomogeneous System Using the Method of Undetermined Coefficients
To solve the given nonhomogeneous system of differential equations using the method of undetermined coefficients, follow the steps below:
The system is given as:
\[ \mathbf{X}' = \begin{pmatrix} 4 & \frac{1}{3} \\ 9 & 6 \end{pmatrix} \mathbf{X} + \begin{pmatrix} -7 \\ 10 \end{pmatrix} e^t \]
where \( \mathbf{X}(t) = \begin{pmatrix} x_1(t) \\ x_2(t) \end{pmatrix} \).
1. **Find the General Solution of the Homogeneous System:**
```math
\mathbf{X}' = \begin{pmatrix} 4 & \frac{1}{3} \\ 9 & 6 \end{pmatrix} \mathbf{X}
```
2. **Determine the Particular Solution:**
Assume a particular solution of the form:
```math
\mathbf{X}_p(t) = \begin{pmatrix} x_{1p}(t) \\ x_{2p}(t) \end{pmatrix} = \begin{pmatrix} A \\ B \end{pmatrix} e^t
```
3. **Substitute the Particular Solution Back into the Nonhomogeneous Equation:**
```math
\begin{pmatrix} x_{1p}' \\ x_{2p}' \end{pmatrix} = \begin{pmatrix} 4 & \frac{1}{3} \\ 9 & 6 \end{pmatrix} \begin{pmatrix} x_{1p} \\ x_{2p} \end{pmatrix} + \begin{pmatrix} -7 \\ 10 \end{pmatrix} e^t
```
Substitute \(\mathbf{X}_p(t) = \begin{pmatrix} A \\ B \end{pmatrix} e^t \) and solve for \( A \) and \( B \).
4. **Combine the General Solution and Particular Solution:**
```math
\mathbf{X}(t) = \mathbf{X}_h(t) + \mathbf{X}_p(t)
```](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F77d2f4c8-4183-4052-877b-b5f84c5df55f%2Fda5425e5-67ae-40f5-9c33-1c5697436fb5%2Fnj8xkot_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving a Nonhomogeneous System Using the Method of Undetermined Coefficients
To solve the given nonhomogeneous system of differential equations using the method of undetermined coefficients, follow the steps below:
The system is given as:
\[ \mathbf{X}' = \begin{pmatrix} 4 & \frac{1}{3} \\ 9 & 6 \end{pmatrix} \mathbf{X} + \begin{pmatrix} -7 \\ 10 \end{pmatrix} e^t \]
where \( \mathbf{X}(t) = \begin{pmatrix} x_1(t) \\ x_2(t) \end{pmatrix} \).
1. **Find the General Solution of the Homogeneous System:**
```math
\mathbf{X}' = \begin{pmatrix} 4 & \frac{1}{3} \\ 9 & 6 \end{pmatrix} \mathbf{X}
```
2. **Determine the Particular Solution:**
Assume a particular solution of the form:
```math
\mathbf{X}_p(t) = \begin{pmatrix} x_{1p}(t) \\ x_{2p}(t) \end{pmatrix} = \begin{pmatrix} A \\ B \end{pmatrix} e^t
```
3. **Substitute the Particular Solution Back into the Nonhomogeneous Equation:**
```math
\begin{pmatrix} x_{1p}' \\ x_{2p}' \end{pmatrix} = \begin{pmatrix} 4 & \frac{1}{3} \\ 9 & 6 \end{pmatrix} \begin{pmatrix} x_{1p} \\ x_{2p} \end{pmatrix} + \begin{pmatrix} -7 \\ 10 \end{pmatrix} e^t
```
Substitute \(\mathbf{X}_p(t) = \begin{pmatrix} A \\ B \end{pmatrix} e^t \) and solve for \( A \) and \( B \).
4. **Combine the General Solution and Particular Solution:**
```math
\mathbf{X}(t) = \mathbf{X}_h(t) + \mathbf{X}_p(t)
```
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