### Solving Systems of Ordinary Differential Equations (ODEs)For the two following systems of ODEs, answer the following questions:#### System I:\[\dot{x}(t) = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} x , \quad x(0) = \begin{pmatrix} 3 \\ 1 \end{pmatrix}\]#### System II:\[\dot{x}(t) = \begin{pmatrix} 5 & -1 \\ 3 & 1 \end{pmatrix} x , \quad x(0) = \begin{pmatrix} 2 \\ -1 \end{pmatrix}\]### Questions:1. **Find the eigenvalues and eigenvectors of matrix A.**2. **Find the general solution (ignoring initial conditions).**3. **Find the particular solution (using the initial condition).**4. **Plot each component of the solution as a function of time.**5. **Plot the vector plot (i.e., on the $x_1, x_2$-plane).**---For both systems, it is essential to determine the eigenvalues and eigenvectors to solve the system analytically. You will use these results to write the general solution. Then, by applying the initial conditions, calculate the particular solution. Graphical representations will help visualize how each component changes over time and provide insight into the behavior of the system.

We have X'(t) = 5-131X., along with the initial condition X(0)= 2-1. The objective here is to find…

Answered: (II) x't) = /5-)* X(@) = 1> Find the…

(II) x't) = /5-)* X(@) = 1> Find the eigenvalues of eigunvecdars of A 3. Z Find the general salution lignoning initial eandition) 32 Find the particular solution (using the initial eandition)

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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