Problem 2. Consider the system of ODEs given by())=(-2)()1. Find the eigenvalues and eigenveectors and obtain the general solution of this systemsof ODES2. Sketch the main features of the phase portrait around the critical point by hand. Hint: Use the relations du = 12dy₁

The objective is to solve the system of ODE's and sketch the phase portrait. y1˙y2˙=4-444y1y2

Answered: Problem 2. Consider the system of ODEs…

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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10. Find the general solution of the system of differential equations 3 -2 -2 d. X = -3 -2 -6 X dt 3 10 1 + 2tet + 3t?et + 4t°et 3 1 -3 Hint: The characteristic polymomial of the coefficient matrix is -(A- 4)²(A- 3). Moreover (:) 2 1 Xp(t) = t²et +t³et +t'e3t -1 -1 -3 is a particular solution of the system.
9. Transform the following equations to a canonical form (i.e. without cross-derivatives): (i) 4Uxx +12Uxy + 9Uyy = 0. (ii) 2Uxx - 4Uxy - 6Uyy + Ux = (iii) Uxx + 2Uxy + 17Uyy = 0. To which type do each of the equations above belongs to? 0.
PDE question
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
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Problem 2. Consider the system of ODEs given by ()-()) 1. Find the eigenvalues and eigenveectors and obtain the general solution of this systems of ODES 2. Sketch the main features of the phase portrait around the critical point by hand. Hint : Use the relations 2 = 2. dy₁