The two given vectors satisfy the given system of linear differential equations. But do they form a fundamental solution set to the system? Please use the Wronskian to calculate The image contains a mathematical problem related to differential equations and fundamental solution sets. Below is a transcription and explanation suitable for an educational website:---**Mathematical Problem:**Consider the following vectors, \( X_1 \) and \( X_2 \):\[ X_1 = \begin{bmatrix} e^{-2t} \\ e^{-2t} \end{bmatrix} \]\[ X_2 = \begin{bmatrix} e^{-5t} \\ -2e^{-5t} \end{bmatrix} \]Define the matrix differential equation:\[ X' = \begin{bmatrix} -3 & 1 \\ 2 & -4 \end{bmatrix} X \]**Question:**Do \( X_1 \) and \( X_2 \) form a fundamental solution set for the given differential equation?**Explanation:**To determine if \( X_1 \) and \( X_2 \) form a fundamental solution set, we need to verify if they are solutions to the differential equation \( X' = AX \) and if they are linearly independent over the interval of interest. This involves checking if the Wronskian of the solutions is non-zero at some point in the interval. The calculations of derivatives and substitution into the differential equation, as well as the computation of the Wronskian, will provide these verifications.*Note: The Wronskian, for functions that are exponentially decaying with different rates, can help in establishing the linear independence provided they solve the system correctly.*--- This layout aims to guide students through the problem and its methodology, making it suitable for an educational website.

For the differential equation, to checkand the set is fundamental set of solution.

Answered: X= [+] -3 1 X × ² = [ 2² -4 ] X…

Math

Advanced Math

X= [+] -3 1 X × ² = [ 2² -4 ] X X2-26-561 est

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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Question

The two given vectors satisfy the given system of linear differential equations. But do they form a fundamental solution set to the system? Please use the Wronskian to calculate

The image contains a mathematical problem related to differential equations and fundamental solution sets. Below is a transcription and explanation suitable for an educational website:

---

**Mathematical Problem:**

Consider the following vectors, \( X_1 \) and \( X_2 \):

\[ X_1 = \begin{bmatrix} e^{-2t} \\ e^{-2t} \end{bmatrix} \]

\[ X_2 = \begin{bmatrix} e^{-5t} \\ -2e^{-5t} \end{bmatrix} \]

Define the matrix differential equation:

\[ X' = \begin{bmatrix} -3 & 1 \\ 2 & -4 \end{bmatrix} X \]

**Question:**

Do \( X_1 \) and \( X_2 \) form a fundamental solution set for the given differential equation?

**Explanation:**

To determine if \( X_1 \) and \( X_2 \) form a fundamental solution set, we need to verify if they are solutions to the differential equation \( X' = AX \) and if they are linearly independent over the interval of interest. This involves checking if the Wronskian of the solutions is non-zero at some point in the interval. The calculations of derivatives and substitution into the differential equation, as well as the computation of the Wronskian, will provide these verifications.

*Note: The Wronskian, for functions that are exponentially decaying with different rates, can help in establishing the linear independence provided they solve the system correctly.*

---

This layout aims to guide students through the problem and its methodology, making it suitable for an educational website.

With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.

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