Wronskian
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
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.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03106fa3-330c-422f-ad56-a6718a5c3a13%2F071fa796-b5c9-4641-8f08-8c5fee0b8dc0%2Fs142t9n_processed.jpeg&w=3840&q=75)
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F03106fa3-330c-422f-ad56-a6718a5c3a13%2F071fa796-b5c9-4641-8f08-8c5fee0b8dc0%2Fp1fkv0g_processed.jpeg&w=3840&q=75)
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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