Verify that the vector 𝐗 is a solution of the given homogeneous linearsystem.𝐗^'=( -1 (1)

step 1 In this question we have a homogenous linear system x prime is equal to minus 1 1 1 into x. We a

Verify that the vector 𝐗 is a solution of the given...

Verify that the vector $\mathbf{X}$ is a solution of the given homogeneous linearsystem.$$\mathbf{X}^{\prime}=\left(\begin{array}{rr}-1 & \frac{1}{4} \\1 & -1\end{array}\right) \mathbf{X} ; \quad\mathbf{X}=\left(\begin{array}{r}-1 \\2\end{array}\right) e^{-3 r / 2}$$

Key Concepts

Verification by Substitution

Verification of a proposed solution involves differentiating the candidate vector solution with respect to the independent variable and substituting both the solution and its derivative into the original differential equation. If both sides of the equation are equal after simplification, the candidate function is confirmed as a valid solution.

Exponential Solutions in Linear Systems

In linear homogeneous systems, solutions are typically expressed in terms of exponential functions, often involving eigenvalues and eigenvectors of the system matrix. The exponential term captures the growth or decay characteristics of the solution, while the constant vector (or eigenvector) indicates the direction in which the system evolves.

Homogeneous Linear Systems of Differential Equations

These systems consist of differential equations where the derivative of a vector function is equal to a matrix acting on that vector, and there are no external forcing terms. It serves as a fundamental model in many applications where the evolution over time is governed solely by the system’s internal dynamics.

Matrix Differential Equations

A matrix differential equation is an equation in which the derivative of a vector is given by the product of a constant or time-dependent matrix and that vector. This framework allows for compact representation of multiple, interdependent linear differential equations.

Transcript

Please give Ace some feedback