Put the eigenvalues in ascending order when you enter x1(t), x2(t) below.T1(t) =8, exp(P t)+8, exp(t)x2(t) =exp(t)+8, exp(8 t) x1(t)Let x(t) =be a solution to the system of differential equations:x2(t)x{(t)x½(t)-23 x1 (t) + 21 x2 (t)-28 x1(t) + 26 x2(t)3If æ(0) =, find æ(t).5

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Answered: x1(t) Let x(t) = be a solution to the…

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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y" +p (t)y' +q(t)y = 0 (i), can be put in more suitable form for finding a solution by making a change of the independent and/or dependent variables. Determine conditions on p and q that enable this equation to be transformed into an equation with constant coefficients by a change of the independent variable. Let r = u (t) be the new independent variable. It is easy to show (島) ( + ( +p(t) + a()y = 0 (1i). dy that dt dz dy d'y and 2 d'y d²z dy dt dz The differential equation becomes dt dr dt d'y d'z de dz dy d'y , 9 and y must be proportional. If g (t) > 0, then choose the constant of proportionality to be 1. Hence, x = u (t) = [g (t)]i dt. With a chosen this way, the coefficient of in equation (ii) is also a constant, provided that the function H = g'(t) +2p(t)q(t) is a dy In order for equation (ii) to have constant coefficients, the coefficients of constant. Thus, equation (i) can be transformed into an equation with constant coefficients by a change of the independent variable,…
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Suppose tx1 + 2x2 + sec(t), sin(t)x1+ tx2 – 3. This system of linear differential equations can be put in the form æ' = P(t)æ + g(t). Determine P(t) and g(t). P(t) = g(t) =
Q(1): Find the general solution of y(4) – 5y" + 4y = 10e-3".
Solve 2 []-[² x(t) = y(t) = 2 -5 -4 X x (0) = 8, y(0) = -10
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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x1(t) Let x(t) = be a solution to the system of differential equations: x2(t) a(t) x½(t) -23 x1 (t) + 21 x2(t) -28 x1(t) + 26 x2(t) 3 If æ(0) = , find æ(t). 5