Suppose(t +4) y {(t - 6) y,==P(t) =8ty₁ + 7y2,3y1 + 8ty2,a. This system of linear differential equations can be put in the formÿ' = P(t)ÿ + ģ(t). Determine P(t) and ģ(t).18ģ(t) =y₁ (1) = 0,y2 (1) = 2.Interval: help (inequalities)[8]b. Is the system homogeneous or nonhomogeneous? Choose↑c. Find the largest interval a < t < b such that a unique solution of the initial valueproblem is guaranteed to exist.

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Answered: Suppose (t + 4) y { (t — 6)y/ - = =…

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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Sometimes it is possible to solve a nonlinear equation by making a change of the dependent variable that converts it into a linear equation. The most important such equation has the form y + p(t)y=q(t)y" and is called Bernoulli's equation after Jakob Bernoulli. If n + 0, 1, then the substitution v-y-* reduces Bernoulli's equation to a linear equation. Solve the given Bernoulli equation by using this substitution. 1²y + 4ty-y²=0,t> 0 +cª +a O O O O O 1+ y = ± 1+ y = ± 5t +c8 22 9 + c +48
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,…
Match the general solutions to the differential equations. ✓ (D² - 18D+81)y=0 (D²+14D+49)y=0 • (D²+16D+64)y=0 (D² - 12D+36) y = 0 A₁y = C₁e6x + C₂ 2te 9x B.y=Ae + Bxe 9x C₁y=Ae-6x + Bxe -7x D.y = Ae + Bxe F. F₁ y=Ae-&r E. y = Ae&x + Bxe &x 8x + Bxe G₁y=Ae-⁹x + Bxe H.y=Ae7x + Bxe 7x 6x -6x -7x -8r -9.x
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
Suppose p and q satisfy the system of differential equations dp dt = p(3 − p+q), dq dt = = q (6 - p - q) ● Starting at (6,2), as t increases will (p, q) approach (5.90, 1.94) or (6.10,2.07)? • Fill in the blanks. The equilibrium points of the system are (0, 0), (3,0), (0,0) and (0.
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