Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Need help for #23

### Differential Equations #### Problem 23 \[ \mathbf{x'} = \begin{bmatrix} 3 & -1 \\ 5 & -3 \end{bmatrix} \mathbf{x}; \, \mathbf{x}_1 = e^{2t} \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \, \mathbf{x}_2 = e^{-2t} \begin{bmatrix} 1 \\ 5 \end{bmatrix} \] #### Problem 24 \[ \mathbf{x'} = \begin{bmatrix} 4 & 1 \\ -2 & 1 \end{bmatrix} \mathbf{x}; \, \mathbf{x}_1 = e^{3t} \begin{bmatrix} -1 \\ -1 \end{bmatrix}, \, \mathbf{x}_2 = e^{2t} \begin{bmatrix} 1 \\ -2 \end{bmatrix} \] #### Problem 25 \[ \mathbf{x'} = \begin{bmatrix} 4 & -3 \\ 6 & -7 \end{bmatrix} \mathbf{x}; \, \mathbf{x}_1 = \begin{bmatrix} 3e^{2t} \\ 2e^{2t} \end{bmatrix}, \, \mathbf{x}_2 = \begin{bmatrix} e^{-5t} \\ 3e^{-5t} \end{bmatrix} \] #### Problem 26 \[ \mathbf{x'} = \begin{bmatrix} 3 & -2 & 0 \\ -1 & 3 & -2 \\ 0 & -1 & 3 \end{bmatrix} \mathbf{x}; \, \mathbf{x}_1 = e^{t} \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}, \] (Note: In Problem 26, the solution for \(\mathbf{x}_2\) is incomplete in the image.)
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Transcribed Image Text:### Differential Equations #### Problem 23 \[ \mathbf{x'} = \begin{bmatrix} 3 & -1 \\ 5 & -3 \end{bmatrix} \mathbf{x}; \, \mathbf{x}_1 = e^{2t} \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \, \mathbf{x}_2 = e^{-2t} \begin{bmatrix} 1 \\ 5 \end{bmatrix} \] #### Problem 24 \[ \mathbf{x'} = \begin{bmatrix} 4 & 1 \\ -2 & 1 \end{bmatrix} \mathbf{x}; \, \mathbf{x}_1 = e^{3t} \begin{bmatrix} -1 \\ -1 \end{bmatrix}, \, \mathbf{x}_2 = e^{2t} \begin{bmatrix} 1 \\ -2 \end{bmatrix} \] #### Problem 25 \[ \mathbf{x'} = \begin{bmatrix} 4 & -3 \\ 6 & -7 \end{bmatrix} \mathbf{x}; \, \mathbf{x}_1 = \begin{bmatrix} 3e^{2t} \\ 2e^{2t} \end{bmatrix}, \, \mathbf{x}_2 = \begin{bmatrix} e^{-5t} \\ 3e^{-5t} \end{bmatrix} \] #### Problem 26 \[ \mathbf{x'} = \begin{bmatrix} 3 & -2 & 0 \\ -1 & 3 & -2 \\ 0 & -1 & 3 \end{bmatrix} \mathbf{x}; \, \mathbf{x}_1 = e^{t} \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}, \] (Note: In Problem 26, the solution for \(\mathbf{x}_2\) is incomplete in the image.)
### Systems of Differential Equations #### Problems 15 to 20 1. **Problem 15:** \[ x' = y + z, \quad y' = z + x, \quad z' = x + y \] 2. **Problem 16:** \[ x' = 2x - 3y, \quad y' = x - 2z, \quad z' = 5y - 7z \] 3. **Problem 17:** \[ x' = 3x - 4y + z + t, \quad y' = x - 3z + t^2, \quad z' = 6y - 7z + t^3 \] 4. **Problem 18:** \[ x' = tx - y + e^tz, \quad y' = 2x + t^2y - z, \quad z' = e^{-t}x + 3ty + t^3z \] 5. **Problem 19:** \[ x_1' = x_2, \quad x_2' = 2x_3, \quad x_3' = 3x_4, \quad x_4' = 4x_1 \] 6. **Problem 20:** \[ x_1' = x_2 + x_3 + 1, \quad x_2' = x_3 + x_4 + t, \quad x_3' = x_1 + x_4 + t^2, \quad x_4' = x_1 + x_2 + t^3 \] #### Instructions for Problems 21 through 30: - First, verify that the given vectors are solutions of the given system. - Use the Wronskian to show that they are linearly independent. - Finally, write the general solution of the system. #### Example Problems 21 and 22: 7. **Problem 21:** \[ x' = \begin{bmatrix} 4 & 2 \\ -3 & -1 \end{bmatrix} x; \quad x_1 = \begin{bmatrix} 2e^t \\ -3e
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Transcribed Image Text:### Systems of Differential Equations #### Problems 15 to 20 1. **Problem 15:** \[ x' = y + z, \quad y' = z + x, \quad z' = x + y \] 2. **Problem 16:** \[ x' = 2x - 3y, \quad y' = x - 2z, \quad z' = 5y - 7z \] 3. **Problem 17:** \[ x' = 3x - 4y + z + t, \quad y' = x - 3z + t^2, \quad z' = 6y - 7z + t^3 \] 4. **Problem 18:** \[ x' = tx - y + e^tz, \quad y' = 2x + t^2y - z, \quad z' = e^{-t}x + 3ty + t^3z \] 5. **Problem 19:** \[ x_1' = x_2, \quad x_2' = 2x_3, \quad x_3' = 3x_4, \quad x_4' = 4x_1 \] 6. **Problem 20:** \[ x_1' = x_2 + x_3 + 1, \quad x_2' = x_3 + x_4 + t, \quad x_3' = x_1 + x_4 + t^2, \quad x_4' = x_1 + x_2 + t^3 \] #### Instructions for Problems 21 through 30: - First, verify that the given vectors are solutions of the given system. - Use the Wronskian to show that they are linearly independent. - Finally, write the general solution of the system. #### Example Problems 21 and 22: 7. **Problem 21:** \[ x' = \begin{bmatrix} 4 & 2 \\ -3 & -1 \end{bmatrix} x; \quad x_1 = \begin{bmatrix} 2e^t \\ -3e
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