![Linear Algebra and Its Applications (5th Edition)](https://www.bartleby.com/isbn_cover_images/9780321982384/9780321982384_largeCoverImage.gif)
Concept explainers
In the study of engineering control of physical systems, a standard set of differential equations is transformed by Laplace transforms into the following system of linear equations;
where A is n × n. B is n × m, C is m × n, and s is a variable. The vector u in ℝm is the “input” to the system, y in ℝm is the “output.” and x in ℝn is the “state” vector. (Actually, the
19. Assumed A– sIn is invertible and view (8) as a system of two matrix equations. Solve the top equation for x and substitute into the bottom equation. The result is an equation of the form W(s)u = y, where W(s) is a matrix that depends on s. W(s) is called the transfer function of the system because it transforms the input u into the output y. Find W(s) and describe how it is related to the partitioned system matrix on the left side of (8). See Exercise 15.
![Check Mark](/static/check-mark.png)
Want to see the full answer?
Check out a sample textbook solution![Blurred answer](/static/blurred-answer.jpg)
Chapter 2 Solutions
Linear Algebra and Its Applications (5th Edition)
- Let x=x(t) be a twice-differentiable function and consider the second order differential equation x+ax+bx=0(11) Show that the change of variables y = x' and z = x allows Equation (11) to be written as a system of two linear differential equations in y and z. Show that the characteristic equation of the system in part (a) is 2+a+b=0.arrow_forwardRewrite the following systems of differential equations as first order systems, written in the standard way (i.e. in the same form as in the previous two problems). To do that, you will need to introduce new unknowns and equations. You do not need to find the solution to the resulting first order systems. a) Rewrite the differential equation as a first order system: y'''-3y''+4y = 3cost b) Rewrite the system of differential equations as a first order system: x1''+ 2x2' + 5x1'= e4tx2''+ 6x1 − x2 = e3tarrow_forwardDetermine a system of first-order differential equations that describes the currents i₂(t) and i3 (t) in the electrical network shown in the figure below. Ai₁ E diz dl3 dt w R₁ + R₁/2 + i₂ R₂ 00000 L2 R3 )/2 + R₁l3 = E )/3 = 3 = Earrow_forward
- Rewrite the system of linear equations [1 -47 x [] = As a single second order differential equation for x x" = 2 x' + Xarrow_forwardWhich of the following are first-order linear equations? y +x?y = 1 xy +y = e (b) y' +xy2 = 1 (d) xy + y = eyarrow_forwardNeed help with this Linear First Order Mixing Problem. Thank you!arrow_forward
- Assume that N(t) denotes the density of an insect species at time t and P(t) denotes the density of its predator at time t. The insect species is an agricultural pest, and its predator is used as a biological control agent. Their dynamics are given below by the system of differential equations. Complete parts (a) through (c). dN = 7N - 5PN dt dP = 4PN - P dt ..... (a) Explain why dN = 7N describes the dynamics of the insect in the absence of the predator. dt If there are no predators present, then P(t) = for all t. Substitute P = in the given differential dN equations to get dt So in the absence of the predators, the above equation describes the dynamics of the insect population. dN Solve the equation, dt N(t) = (Type an expression using t as the variable.) Describe what happens to the insect population in the absence of the predator. In the absence of the predator, the insect populationarrow_forwardDo d and e Please write outarrow_forwardConsider the second order differential equation as below:2?̈+ 12?̇ = 10?a) Change the above equation into matrix form of ?̇ = ??b) Write the quadratic form related to matrix Aarrow_forward
- Consider the SIR model for disease control and prevention measures: immunization and quarantine pS bN S I R ds IP dR Figure 1: Transfer diagram for an SIR Vaccination model. i. Derive the system of differential equations for the model with transfer diagram for S(0) So, I(0) = lo = 0 and R(0) = Ro = 0. ii. Explain the physical meaning of each of the parameters p, b, 2,y, d in each compartment iii. State the Ro value for this model and explain what it represents iv. Find the disease free equilibrium point v. Find the Jacobian analyse the stability of the disease free equilibrium point vi. Find an endemic equilibrium pointarrow_forwardConsider y′′′ −y′′ +4y′ −4y = 0 a) Convert to a matrix differential equation X′ = AX where A is a 3 x 3 matrix. b) Show that one solution is et. Find the other solutions and put in vector form.arrow_forwardExercise 6.7.8 Classify the equilibria of the following linear differential equations: [X'=Y Y' = -2X-3Y a) b) X' = 4x +3Y Y' = X-2Yarrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningElements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,
![Text book image](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
![Text book image](https://www.bartleby.com/isbn_cover_images/9781285463230/9781285463230_smallCoverImage.gif)