The system of first order differential equations:has solution:Yı(t)Y2 (t)=1/2==3 y₁ + 2y2,-34 y₁ - 13 y2,Note: You must express the answer in terms of real numbers only.31 (0) = 432 (0) = -20

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Answered: The system of first order differential…

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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Solve the system of differential equations. - 6x1 + 2x2 X1 X₂' = - 2x1 - 10x₂ x₁ (0) = – 2, x₂(0) = - 22 x₁ (t) = -22e -22e-8t +48e t(-+- 2/2 ) x₂ (t) = −22e-8t+48e-8t(t) - X OT
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
Consider the system of differential equations x₁ = 10/3x₁ +4/3x2 x2 = 8/3x1 +14/3x2 9 where 1 and 2 are functions of t. Our goal is first to find the general solution of this system and then a particular solution.
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 describing the progress of a disease in a population, given by In our particular case, this is: a) Find the nullclines (simplest form) of this system of differential equations. The x-nullcline is y = x = 3-3xy - 1x y = 3xy - 2y where x (t) is the number of susceptible individuals at time t and y(t) is the number of infected individuals at time t. The number of individuals is counted in units of 1,000 individuals. The y-nullclines are y = and x = b) There are two equilibrium points that are biologically meaningful (i.e. whose coordinates are non-negative). They are (x1,9₁) and (x2, y2), where we order them so that x1 < x2. The equilibrium with the smaller x-coordinate is (x₁, y₁) : The equilibrium with the larger x-coordinate is (x2, y₂) = where A = 0 sin (a) c) The linearization of the system of differential equations at the equilibrium (ï1, Y₁) gives a system of the form (+) ¹ (~), f əx ∞ a Ω = A x' G E = F(x, y) for a vector-valued function…
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The system of first order differential equations: y/2 = = 3 y₁ + 2y2, -34 yı 13 y2, - 3₁ (0) = 4 32 (0) = -20