Show that the given functions are solutions of the system x'(t) = A(x)x(1) for the given matrix A. Then use the Wronskian to show that they are linearly independent. Finally, write the general solutions. 3e [ 2e² />*, (t) = x; (1) = e 3eSt [4 -3 A = 6. -7
Show that the given functions are solutions of the system x'(t) = A(x)x(1) for the given matrix A. Then use the Wronskian to show that they are linearly independent. Finally, write the general solutions. 3e [ 2e² />*, (t) = x; (1) = e 3eSt [4 -3 A = 6. -7
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter3: Determinants
Section3.CM: Cumulative Review
Problem 17CM: Find the sequence of the elementary matrices whose product is the non singular matrix below. [2410]
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![**Problem Statement:**
3. Show that the given functions are solutions of the system \( x'(t) = A x(t) \) for the given matrix \( A \). Then use the Wronskian to show that they are linearly independent. Finally, write the general solutions.
**Functions:**
\[ x_1(t) = \begin{bmatrix} 3e^{2t} \\ 2e^{2t} \end{bmatrix}, \quad x_2(t) = \begin{bmatrix} e^{-5t} \\ 3e^{-5t} \end{bmatrix} \]
**Matrix:**
\[ A = \begin{bmatrix} 4 & -3 \\ 6 & -7 \end{bmatrix} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F86847a45-621f-4eb2-b143-0cde4fac410f%2Fc0d97483-484e-45e7-88a7-df62949b65a1%2Fbne5xxi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
3. Show that the given functions are solutions of the system \( x'(t) = A x(t) \) for the given matrix \( A \). Then use the Wronskian to show that they are linearly independent. Finally, write the general solutions.
**Functions:**
\[ x_1(t) = \begin{bmatrix} 3e^{2t} \\ 2e^{2t} \end{bmatrix}, \quad x_2(t) = \begin{bmatrix} e^{-5t} \\ 3e^{-5t} \end{bmatrix} \]
**Matrix:**
\[ A = \begin{bmatrix} 4 & -3 \\ 6 & -7 \end{bmatrix} \]
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