Consider a system of two linear first-order ordinary differential equations:y₁ = y₁ - y2, y₂ = 2y1 - y2 -a) The correspondingOλ₁ = 1, №₂ = −1b) The correspondingOIII and IVwhere1:4₁=II:42=1+2(¹+4)(¹3²)1-2i2(1 + i)(2(1²-+-+))i)III:u1IV:4₂ =eigenvalues are=OA₁ = 1 + 1, A₂ = 1 - iOA₁i, A₂ = −i=eigenvectors of this linear ODE system are:Oll and IIIOI and IVOI and IIc) The phase portrait for this system of ODEs isStable focus with spiral in Centre Unstable focus with spiral out Stable node

Given system of differential equationsy˙1=y1−y2y˙2=2y1−y2In Matrix form [y˙1y˙2]=[1−12−1][y1y2]Let…

Answered: Consider a system of two linear…

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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Consider the second-order equation d 2y/dt 2 + p dy/dt +qy = 0 a. If p=q=1, compute the eigenvalues and eigenvectors of the coefficient matrix. What would be the long-term behavior of the solutions in this case? b. If p=0 and q=2 determine the general solution.
a. Find the eigenvalues and eigenvectors of the matrix x₁ (t) x₂ (t) = X₁ b. Solve the system of differential equations a' = = = V₁ = [₁ 5 -3 4 -2 and X₂ = satisfying the initial conditions V₂ = [₁8] =[3]
Consider a dynamic system described by the following integro-differential equation t 4*+3x+2x+5|x(t)dt = 6r +i 0. Derive a state representation for the system, with r as the input and y = 2x+i as the output.
[3 [4 Q2A)Matrix A = = [(2² 21₁B =[ ²₂ ] B = [₁2₂], find AB. 2. dy B)Solve differential equation dx -(2ln5y+x-¹) 2xy-¹
Read each of the two statements carefully. Choose A(if both statements are true) , B(if the 1st statement is true but the 2nd is false), c(if the 1st statement is false and the 2nd is true and D(if both statements are false. 1st: dx2 =0 is a 2nd order linear partial differential equation ду? + 2nd : dx? =0 is a 1st degree partial differential equation ду2 A) D В) в A D) c
Consider the second-order system of differential equations dy d²x dt² =x-5y, The coefficient matrix 1 -5 eigenvalues are 6 and -4. Select the option that gives the general solution of this system. I dt² C₁ -5x + y. 1 -5 11 1 has eigenvectors Select one: [*] = C₁ [ ¹₁ ] cos(√õt) + C₂ [ ¹₁ ] sin(√6t) + Cs [1] cos(2t) + C₁ [1] C3 C4 sin(2t) evot + Vot C3 - [3] = a₁ [¹₁] ev [*] = C₁ [1] cos(√ēt) + C₂ H ₂[1] HHHHH [+] = C₁₂ [ ¹₁ ] ev√² + C₂ [ ¹₁ ] e-√₂ 3 [1] 2²¹ -C₂₁ [ ²₁ ] e-√² + C₂ [1] sin(√6t) + C3 [1₁] [1] Cos evot +03 e2t+CA C₂ [4] [1] and H cos(√6t) + C₂ and the corresponding cos(2t) + C4 H e2t + C4 sin(2t) -2t [¹1] e ²2 -2t e2t sin (√6t) + C3 as He" + a₁₂H₁² -2t
a). consider the system of differential equations. 21 5²²lt) = [ ²1 ] ²lt), where Rit) = [kilt)] 23 Find the general solution to this system, and write out the solutions for Xilt) and X₂lt), without using vector notation.
Consider a system of differential equations which has the general solution: 18]- = C, · cos (5. t) – sin (5 - t) )+C2 · sin (5 - t) + • Cos (5 - t (a) Determine the fundamental matrix (t) which has the property (0) = I. (t) = (b) Use the fundamental matrix found in part (a) to quickly determine the particular solution for the following initial conditions: r (t) : y (t) = :8]-[- (0): 1 (ii) r (t) = y (t) =
I already know the eigenvalues are -2+6i, and 2-6i, and the eigenvector is [6i,1] , and [-6i,1], im just struggling finding the overall solution
1b
Solve the following system of linear differential equations: y = 6y, - y2 y, = 2y, + 3y, а. 1 1 5r = k,e 2 4r + kge be 2 y 1 5r = k,e + kze² = k,e -4r + ke -5r Od. 1 -41 + ke -5r = kje
dy dy +3 dr? 6. Consider the second-order equation - 10y = 0. di (a) Convert the equation to a first-order, linear system. (b) Compute the eigenvalues and eigenvectors of the system. (c) For each eigenvalue, pick an associated eigenvector V, and determine the solution Y(t) to the system. (d) Compare the results of your calculation with the results that you obtained when you used the guess-and-test method in Homework 6, question 4.

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