Apply the method of undetermined coefficients to find a particular solution to the following system. x' = 5x - 7y + 12, y'=x-3y -4e-2 2t Xp (t) = - 2t 2t 818-8 te

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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**Application of the Method of Undetermined Coefficients**

To find a particular solution to the following system, apply the method of undetermined coefficients:

\[
x' = 5x - 7y + 12, \quad y' = x - 3y - 4e^{-2t}
\]

**Particular Solution Expression**

The particular solution \( x_p(t) \) is given by:

\[
x_p(t) = \begin{bmatrix} 1 \\ 1 \end{bmatrix} te^{-2t} - \begin{bmatrix} 1 \\ 0 \end{bmatrix} e^{-2t} - \begin{bmatrix} \text{[missing]} \\ \text{[missing]} \end{bmatrix}
\]

**Explanation of Components:**

1. **Matrix Components:**

   - The first matrix \( \begin{bmatrix} 1 \\ 1 \end{bmatrix} \) is multiplied by \( te^{-2t} \), representing a solution that involves time-dependent exponential decay with a linear multiplier \( t \).

   - The second matrix \( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \) is multiplied by \( e^{-2t} \), indicating an exponential decay solution component.

2. **Missing Matrix:**

   - The placeholders \([missing]\) suggest that there may be additional components or steps in the solution not shown in this snippet. The full solution would require completing these steps based on the equations and method used.

This structure is part of the undetermined coefficients method, where you hypothesize a particular solution form and adjust the coefficients to satisfy the system of equations.
Transcribed Image Text:**Application of the Method of Undetermined Coefficients** To find a particular solution to the following system, apply the method of undetermined coefficients: \[ x' = 5x - 7y + 12, \quad y' = x - 3y - 4e^{-2t} \] **Particular Solution Expression** The particular solution \( x_p(t) \) is given by: \[ x_p(t) = \begin{bmatrix} 1 \\ 1 \end{bmatrix} te^{-2t} - \begin{bmatrix} 1 \\ 0 \end{bmatrix} e^{-2t} - \begin{bmatrix} \text{[missing]} \\ \text{[missing]} \end{bmatrix} \] **Explanation of Components:** 1. **Matrix Components:** - The first matrix \( \begin{bmatrix} 1 \\ 1 \end{bmatrix} \) is multiplied by \( te^{-2t} \), representing a solution that involves time-dependent exponential decay with a linear multiplier \( t \). - The second matrix \( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \) is multiplied by \( e^{-2t} \), indicating an exponential decay solution component. 2. **Missing Matrix:** - The placeholders \([missing]\) suggest that there may be additional components or steps in the solution not shown in this snippet. The full solution would require completing these steps based on the equations and method used. This structure is part of the undetermined coefficients method, where you hypothesize a particular solution form and adjust the coefficients to satisfy the system of equations.
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