Consider the second-order system of differential equationsdyd²xdt²=x-5y,The coefficient matrix1-5eigenvalues are 6 and -4.Select the option that gives the general solution of this system.Idt²C₁-5x + y.1-5111has eigenvectorsSelect one:[*] = C₁ [ ¹₁ ] cos(√õt) + C₂ [ ¹₁ ] sin(√6t) + Cs [1] cos(2t) + C₁ [1]C3C4sin(2t)evot+VotC3-[3] = a₁ [¹₁] ev[*] = C₁ [1] cos(√ēt) + C₂H 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] Cosevot+03e2t+CAC₂ [4][1] and Hcos(√6t) + C₂and the correspondingcos(2t) + C4He2t + C4sin(2t)-2t[¹1] e ²2-2te2tsin (√6t) + C3as He" + a₁₂H₁²-2t

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Answered: Consider the second-order system of…

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 initial value problem '=[¯6 __!];, ;(0) = (0) = [₂] a. Find the eigenvalue A, an eigenvector ₁, and a generalized eigenvector ₂ for the coefficient matrix of this linear system. λ = .v₁ = -1 b. Find the most general real-valued solution to the linear system of differential equations. Use I as the independent variable in your answers. y(t) = C₁ +0₂ c. Solve the original initial value problem. y₁ (1) = y₂ (1) =
A 3×3 real matrix has three eigenvalues. One of them is λ₁ = Another is A₂ = 1 + i with eigenvector v2 - (3) = -1 with eigenvector v₁ = (a) What is the third eigenvalue and its corresponding eigenvector? (b) Find the general solution to x' = Ax, in purely real form. (c) Find the solution to x' = Ax subject to initial condition x (0) = (:). (6) 1
ÿ' = -[3. a. Find the eigenvalues and eigenvectors for the coefficient matrix. i 3/1-1/16 5 5 A₁ = i = 1 • Find the real-valued solution to the initial value problem yí = ly/₂ = Use t as the independent variable in your answers. 31(t) = 11 y2(t) = -15 15 2 -5 -3 3y1 + 2y2, -5y1 - 3y2, y. and A₂ = -i 9 y₁ (0) = 11, Y2(0) = −15. 9 V2 = T 1 i 3/4 + 1/4 5 35
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.
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
lloa [Ur ses SEC::116 Question 9 Find the general solution to the system x' = Ax where A is the given matrix. -2 -1 1 A= -1 -9 3 -1 -2 -4 Hint: -6 is an eigenvalue.
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
Q1) Solve the following system of algebraic equations using matrix inverse method. 6 30-0 Q2/ Solve the following differential equations. 1) + y² - 4: 2) 7+3 / +2y=0 Q3/ Find points of intersection for the following curves. 1- r = 1+ cos 0, 2- r = 1 + sin 0.
Q7

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