Solve the system of differential equations given in matrix form for the general solution. We must do this by calculating the eigenvalues and eigenvectors. Note: we get a repeated eigenvalue and must solve for the second eigenvector to get two linearly independent solutions for our general solution. Please explain how you calculate the second eigenvector The image contains a mathematical representation of a system of differential equations. The notation used is typical in linear algebra and differential equations.The system of equations is given by:\[ z' = \begin{bmatrix} -1 & 3 \\ -3 & 5 \end{bmatrix} z, \quad z(0) = z_0 \]This represents a first-order linear system, where \( z \) is a vector function of a variable, typically time.- **\( z \) is defined as**: \[ z = \begin{bmatrix} z_1 \\ z_2 \end{bmatrix} \]- **\( z' \) is the derivative of \( z \) with respect to time, defined as**: \[ z' = \begin{bmatrix} z_1' \\ z_2' \end{bmatrix} \]The initial condition for the system is \( z(0) = z_0 \), indicating the state of the system at time \( t = 0 \).This kind of system is typically solved to understand the behavior of dynamic systems in various fields such as physics, engineering, and economics.

Given that the system of the differential equations i.e. Where To find the general solution to the given system of differential equation.Now, we have Compare the above equation with the following equation i.e. Here, For Eigenvalues and Eigenvectors: So are the eigenvalues of the matrix A.For eigenvectors: Let Subtituting these values in the given system of differential equation and we obtain.By comparing like terms and we get. Therefore, The homogeneous solution to the given differential equation is: By using the given condition i.e. in the above equation and we get.Therefore, Hence the required general solution to the given system is

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olve the system of differential equations given in matrix form for the general solution. We must do this by calculating the eigenvalues and eigenve

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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Question

Solve the system of differential equations given in matrix form for the general solution. We must do this by calculating the eigenvalues and eigenvectors. Note: we get a repeated eigenvalue and must solve for the second eigenvector to get two linearly independent solutions for our general solution. Please explain how you calculate the second eigenvector

The image contains a mathematical representation of a system of differential equations. The notation used is typical in linear algebra and differential equations.

The system of equations is given by:

\[ z' = \begin{bmatrix} -1 & 3 \\ -3 & 5 \end{bmatrix} z, \quad z(0) = z_0 \]

This represents a first-order linear system, where \( z \) is a vector function of a variable, typically time.

- **\( z \) is defined as**:
\[ z = \begin{bmatrix} z_1 \\ z_2 \end{bmatrix} \]

- **\( z' \) is the derivative of \( z \) with respect to time, defined as**:
\[ z' = \begin{bmatrix} z_1' \\ z_2' \end{bmatrix} \]

The initial condition for the system is \( z(0) = z_0 \), indicating the state of the system at time \( t = 0 \).

This kind of system is typically solved to understand the behavior of dynamic systems in various fields such as physics, engineering, and economics.

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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