= y2,Consider a system of two linear first-order ordinary differential equations: y₁a) The corresponding eigenvalues areA₁ = 3, 1₂ = −3b) The corresponding eigenvectors of this linear ODE system areI and IVII and IIIwhereI:u₂ =II:u₂III:u₁IV:u₁(3₁)(-3₁)(3)= (³1)3i3i==λ₁ = 3i, λ₂ = -3ic) The phase portrait for this system of ODEs isUnstable focus with spiral outI and IIIStable focus with spiral iny2 = -9y₁.₁9, λ₂ = -9I and IIIStable nodeCentre

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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 a system of two linear first-order ordinary differential equations: y₁ = y₁ - y2, y₂ = 2y1 - y2 - a) The corresponding Oλ₁ = 1, №₂ = −1 b) The corresponding OIII and IV where 1:4₁ = II:42 = 1+ 2 (¹+4) (¹3²) 1 - 2i 2(1 + i) (2(1²-+-+)) i) III:u1 IV:4₂ = eigenvalues are = OA₁ = 1 + 1, A₂ = 1 - i OA₁i, A₂ = −i = eigenvectors of this linear ODE system are: Oll and III OI and IV OI and II c) The phase portrait for this system of ODEs is Stable focus with spiral in Centre Unstable focus with spiral out Stable node
(2 cos 2t / 19. Assume that a real 2 x 2 matrix A has an eigenvalue of A1 = 3+ 2i with a corresponding eigenvector = (6) Find the general solution of the system of differential equations 2i given by 글 %=D Az. cos 2t sin 2t 교(t) = cie3t ( sin 2t + cze³t 2 cos 2t cos 2t sin 2t a(t) = c1e3t + cze3t -2 sin 2t 2 cos 2t (. cos 2t sin 2t a(t) Ciešt + cze3t sin 2t - cos 2t ( -2 cos 2t sin 2t a(t) = cie3t + cze3t sin 2t -2 cos 2t cos 2t ( 2 sin 2t ) sin 2t a(t) = c1e3t + cze³t 2 cos 2t
For the system of differential equations, 3 23 84 6 - 22 4et a) Find the characteristic polynomial of the matrix of coefficients A. CA(X) A1, A2 b) Find the eigenvalues of A. Enter the eigenvalues as a list in ascending order separated by commas. U₁ = U₂ = c) Find the eigenvectors assuming u₁ is the eigenvector associated with the smaller eigenvalue X₁ and u₂ is the eigenvector associated with the larger eigenvalue λ₂. Enter the eigenvectors as a matrix with an appropriate size. y' = v(t) y + d) Determine a general solution to the system by completing the following steps. i. Find v(t) = [Y-¹(t)f(t)dt . ii. Find a particular solution y(t). H yp(t)= y(t) ii. Then a general solution for the system in the matrix form is -8:33:
Consider the second-order system of differential equations d²x dt² d²y dt2 =x+3y, The coefficient matrix 1 3 3 3x+y. wwwwww has eigenvectors Hand [1] are 4 and - 2. Select the option that gives the general solution of this system. Select one: ° 1) = 0[¹] con (24) + C₁ [₁] (²0₁₁+₂ [1¹] CA cos(2t) sin(2t) + C3 ev2t [*] = C₁ [1] cos(2t) + 0₂ sin(2t) + C3 cos(√2t) + C4 Casin(√21) = + HHH] co √21) + C [¹] sin (√2) C₁ e2t cos(√2t) C4 C3 and the corresponding eigenvalues [3] = C₁ [1] cos(2t) + C₂ e2t 2t [1] = ₂ [1] ² + ₂ [1] ² ² + ₁ [₁¹] e√²+ + C₁ [ 1¹ ] e-√₂² 2t e C4 A WEW MEXAND sin(2t) + C3 · [₁¹] ₁ √²+ + 0₁ [1¹] ₁ = √² √√2t e
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
Solve the following differential system: ¤(0) y(0) 2 with 3 Hint. One eigenvalue and its corresponding eigenvector of A are r = -2+i, uį = r(t) 24 cost ae Then you obtain -2e 4(cost + sin t) Please type in a :
Suppose that a 2 x 2 matrix A with real entries has an eigenvalue X = 1 + 3i with corresponding 4 i [12] eigenvector 1 + 2i Which of the following is NOT a solution of the linear system y' = Ay? y(t) = et y(t) = et y(t) = et y(t) = et 4 cos(3t) sin(3t) [cos(3t) + 2 sin(3t). 4 cos(3t) + sin(3t) cos(3t) 2 sin (3t) - cos(3t) + 4 sin(3t) 2 cos(3t) + sin(3t) cos (3t) - 4 sin (3t) -2 cos(3t) - sin(3t)
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) =

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