solve the following system of linear non-homogeneous differential equations with eigenvalues/eigenvectors and variation of parameters The image presents a system of differential equations in matrix form. It is expressed as follows:\[ \frac{d\mathbf{z}}{d\mathbf{t}} = \begin{bmatrix} 0 & 2 \\ -1 & 3 \end{bmatrix} \mathbf{z} + \begin{bmatrix} 2 \\ e^{-3t} \end{bmatrix} \]where \[\frac{d\mathbf{z}}{d\mathbf{t}} = \begin{bmatrix} z_1' \\ z_2' \end{bmatrix} \]and \[\mathbf{z} = \begin{bmatrix} z_1 \\ z_2 \end{bmatrix} \]This system describes how the vector \(\mathbf{z}\), consisting of the components \(z_1\) and \(z_2\), evolves over time \(t\). The matrix on the right-hand side contains constant coefficients, determining how these components interact. The additional vector represents external input or forcing terms, with an exponential term \(e^{-3t}\) indicating a time-dependent influence.

Answered: Image /qna-images/answer/8316e925-53df-4e76-84ff-4e95610add4e.jpg

Answered: f 02 3 mere dz zi #-[21]) Z + N N 2…

Math

Advanced Math

f 02 3 mere dz zi #-[21]) Z + N N 2 e*3% and Z=

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: ty' + 2y = t²t+1

Q: Determine the ey of the system of linear invowing de (LP differential equations temperature,…

Q: A system of first-order differential equation consists of: y₁ = 3y₁ + 5y2, and y2 = -y₁-3y2- (1)…

Q: Transform the differential equation y" - 2y" +8y-y=t into a system of first order equations. 1₁ =…

Q: 2. x' = Find the general solution of the system of equations. -20 28 8 -14 22 4 X -21 21 14

A: Find: the general solution of the system of equations. x'=-20288-14224-212114x

Q: Consider the system of differential equations a. Rewrite this system as a matrix equation ÿ' = Aỹ.…

Q: x+AX=0, X (6)=0, strum - Liouville paoblem. x (x)=0 comesponding. c) FoN and eigen functions eigen…

A: Consider x''+λx=0with x(0)=0, x(π)=0The characteristics equation is m2+λ=0(i) λ=0⇒m2=0⇒m=0, 0General…

Q: Find eigenvalues and eigenfunctions. Verify the orthogonality of the eigenfunctions. x2y" - xy' + (1…

Q: Suppose the size of a population evolves according to the logistic equa- tion: dN = 1.5N dt N 100…

Q: Using Elgenvalues and Eigenvectors Dx/ dt= y Dy/dt=2x

A: In this question, we find the eigenvalues and eigenvectors for the system of the differential…

Q: Consider the system dx · (1) dt 3 (a) Let x= ge" be the solution of the system (1) then, 5-r -1 (2)…

Q: post office sells 4-cent sheep stamps and 2-cent sunset stamps. Cole bought 5 stamps spent exactly…

Q: Sketch several solution curves in the phase plane of the system of using the given eigenvalues and…

A: The general solution of the given system of DE using the given data is

Q: Calculate the general solution for the homogeneous system of first-order equations. y₁ =6y₁ + y₂ y₂…

A: [Note : As per the guidelines I am answering only one question at a time]Given differential…

Q: Find the Lie point symmetries of the second-order partial differential equation UttUxx+u = 0.

A: Given PDE: The given second-order partial differential equation is , where U is a function of and…

Q: A. y" – 3y' +2y = 0 B. y" + 2y'+ 2y = 0 C. y" +4y = 0 D. y" + 2y' + y = 0 Table 1: A-D are…

A: Here we find roots of characteristic equation and then write the solution using the roots of…

Q: 27. (Critical point) What kind of critical point does y' = Ay have if A has the eigenvalues -6 and…

Q: We can find another solution to a system of differential equations with repeated eigenvalues by…

A: Statement is false. we use variation of parameters to find a particular solution of non homogeneous…

Q: Let X = {a, b, c, d} and TX = {ø, X, {b}, %3D {b,c}, {a,b,d}} be a topology on X. Let Y = {1, 2, 3,…

A: A function is continuous if and only if inverse image of every open set is open

Q: For the nonlinear autonomous system x' = y? + 2y , y' = x – y

A: Given nonlinear autonomous system x'=y2+2yy'=x-y For critical point put y'=0 and x'=0.

Q: Solve the system of differential equations x' = 31x + 56y ly' 16x 29y - - x(0) = 7, y(0) = - 4 The…

Q: Use Eigenvalue method to find the solution of the given linear system: x' = 6x - 4y, y' = x + 2y,…

Q: If the system of differential equation has a chractristic equation 12 – 41 +4 = 0, then the general…

Q: Find eigenvalues associated with this system of differential equations. X. -2 -1 and -2 -2 and -3 O…

Q: Consider the nonlinear system of differential equations dax=(2²+1) (-36-24), 1/1 = (y+¹) (2y=x²-2x)…

Q: x=12²2 ²1x X find the eigenvalues and eigenvectors -Classify the critical point (0.0) as to type

Q: X' = -1 1 12 03 3 0 1 X -1/

A: Consider the provided question, x' =-11012103-1xFor a system of differential equations x'=Ax, assume…

Q: 3. dx/dt = x + y², dy/dt = x + 2y Transcribed Image Text: 3-dx/dt = x+y², dy/dt = x + 2y (a)…

A: Given, the system of differential equation is dxdt = x + y2dydt = x + 2y Let fx, y = x + y2gx, y = x…

Q: Consider the differential equation бу" + 7у" + 5у' — 8у %3D 4u" + би' — Зи Find the input matrix of…

A: Solution:-

Q: Suppose that the differential equation x' = Ax is such that A is a 2 x 2 matrix with eigenvalues A1…

A: Draw the phase potrait : Draw the eigen vectors corresponding to eigen values and corresponding…

Question

100%

solve the following system of linear non-homogeneous differential equations with eigenvalues/eigenvectors and variation of parameters

The image presents a system of differential equations in matrix form. It is expressed as follows:

\[
\frac{d\mathbf{z}}{d\mathbf{t}} = \begin{bmatrix} 0 & 2 \\ -1 & 3 \end{bmatrix} \mathbf{z} + \begin{bmatrix} 2 \\ e^{-3t} \end{bmatrix}
\]

where

\[
\frac{d\mathbf{z}}{d\mathbf{t}} = \begin{bmatrix} z_1' \\ z_2' \end{bmatrix}
\]

and

\[
\mathbf{z} = \begin{bmatrix} z_1 \\ z_2 \end{bmatrix}
\]

This system describes how the vector \(\mathbf{z}\), consisting of the components \(z_1\) and \(z_2\), evolves over time \(t\). The matrix on the right-hand side contains constant coefficients, determining how these components interact. The additional vector represents external input or forcing terms, with an exponential term \(e^{-3t}\) indicating a time-dependent influence.

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.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1: Find general solution.

VIEW

Step 2: Continue...

VIEW

Solution

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

Consider the system of differential equations For this system, the smaller eigenvalue is (-39/10) [Note-- you may want to view a phase plane plot (right click to open in a new window).] If y'= Ay is a differential equation, how would the solution curves behave? • All of the solutions curves would converge towards 0. (Stable node) All of the solution curves would run away from 0. (Unstable node) and the larger eigenvalue is The solution curves would race towards zero and then veer away towards infinity. (Saddle) The solution curves converge to different points. The solution to the above differential equation with initial values (0) = 4, y(0) = 3 is (t)= (49/12)*e^((-9/10)*t)-(1/12)*e^((-19/10)*t) y(t) = (49/12)*e^((-9/10)*t)+(5/12)*e^((-19/10)*t) da dt dy dt = -1.4x +0.75y, = 1.66667 3.4y. (-9/10)
Use a calculator for: - Finding eigenvalues and eigenvectors. - Matrix multiplication still required to show how you find generalized - Matrix inversion - Row reduction -Integration5.
Show that all separable differential equations are exact differential equations.
The behaviour of an unforced mechanical system is governed by the differential equation 5 2 -1 0 x(1) = 3 6-9 x(t), x(0) = 1 1 1 1 0 (a) Show that the eigenvalues of the system matrix are 6, 3, 3 and that there is only one linearly independent eigenvector corresponding to the eigenvalue 3. Obtain the eigenvectors corresponding to the eigenvalues 6 and 3 and a further generalized eigenvector for the eigenvalue 3. (b) Write down a generalized modal matrix M and confirm that AM = MJ for an appropriate Jordan matrix J. x(t) = Me M-¹x(0) obtain the solution to the given differential equation. (c) Using the result
Describe any one of the physical systems where we can apply Legendre or Bessel differential equations.
Consider the system of differential equations = (-3/2 3/2)x+ (-¹1) ¹ x' = -3/2 -1 where two eigenvalues of 2 3/2 -3/2 -1 are r = 1/2 and r₂ = 1/2, and the corresponding eigenvector is v= -(2/3). Let x = Ty where T is a transformation matrix. Find two first order differential equations, eigenvector is u=== and = -/+nte, n =+², ₂ -+-+ M and the generalized
Show your work
Solve the system of ODES given by x' x. Sketch a graph of the eigenvectors and plot 13 some sample trajectories. Is the origin an attractor, a repeller or a saddle point? Give your final solution in exact form with e.