Given that ₁X₁=X2-6x(t)2y(t) ===[¹] andHhelp (numbers)Find the solution to the linear system of differential equationsx' = 6xy = - 4x + 2yand v₂ =v2->←Isatisfying the initial conditions (0) = 2 and y(0) = -1.help (formulas)help (formulas)0- [9]help (numbers)→are eigenvectors of the matrix→6[14 ๆ-4 29determine the corresponding eigenvalues.

Answered: Image /qna-images/answer/87e701f1-402f-4bce-9a06-0daf1aa3345b.jpg

Answered: Given that ₁ A₁ = 6 = H -A help…

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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This is the first part of a two-part problem. Let P=[-: 1 5₁(t) = [(41) 5₂(t) = - sin(4t). a. Show that y₁ (t) is a solution to the system ÿ' = Pÿ by evaluating derivatives and the matrix product y(t) = = 0 [1] -4 Enter your answers in terms of the variable t. -4 sin(4t) -4 cos(4t)] ÿ₁ (t) [181-18] b. Show that y₂ (t) is a solution to the system ÿ' = Pÿ by evaluating derivatives and the matrix product Enter your answers in terms of the variable t. 04] 32(t) = [-28]|2(t) 181-181
1. (a) What are A, X and x' in matrix form x' = AX of the system of differential equations { FO (b) Find all eigenvalues and eigenvectors of A. (c) Find exponential solutions of the system of differential equations. (d) Verify your solutions. x₁ = 3x1 x2 = 5x1 + 2? x2
The system of first order differential equations has solution y₁ (t) = = y₂(t) = 3/₁ ly/2 = 4 y₁ + 1 y2, -6 y₁ - 1 y2, y₁ (0) = 10 Y₂ (0) = -10
Solve the system of differential equations. x' = 2x + 2y y' = - 17x + - 4y x (0) — 2, y(0) = x(t) = y(t) = = - 17 -
The system of first order differential equations has solution y₁ (t) = y₂ (t) = Sy/₁ 13/12 = -7 y₁ - 4 y2, 8 y₁ + 5y2, y₁ (0) Y₂ (0) = -1 = 8
where A system of system of first-order linear differential equations has the form y' = Ay y' = , y = Y1 Y2 , A = a11 a21 This gives us the system B [an an2 ann] Solve the system of linear differential equations by following the steps listed. y₁ = y₁ 432 y₂ = -2y1 + 8y2 a12 022 ain a2n ⠀ (a) We first want to write this as a matrix equation y'= Ay. List out what the matrix A should be in this matrix equation. (b) We would like to have the equations so that y is a function of only y₁ and ₂2 is a function of only y2. To do this, we want to turn A into a diagonal matrix. This can be done by taking the diagonalization of A. Diagonalize the matrix A you created in part a. (c) Now that we have a matrix P such that P-¹AP = D is diagonal, we substitute y' and y to be Pw' = y Pw=y Pw = APw Since P is invertible, we can multiply P-1 to the left on both sides of the equations to get w = P-¹APW = Dw Write out what the system of linear differential equations looks like now. (d) Let k be any scalar.…
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) =
1. Find real-valued fundamental solutions of the differential equation x' = Ax, where matrix A has eigenvalues and eigenvectors - [₁4] - 6.2) A₁ = −2+i V₁ = A2 = -2-i V2 =
which is the solution vector x(t)=(x(t)) of differential equation system (y(t))
Consider the following matrix A = 5 7 0 -2 (a) Find the eigenvalues and the eigenvectors of this matrix. (You need to show how you find them.) (b) Using the eigenvalues and the eigenvectors of A, solve the system of differential equations ' = AT.
can someone explain this for me, please ?

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