Determine the values of a, if any, for which all solutions of thedifferential equationy" – (2a – 14)y' + (a² – 14a + 40)y = 0→ 00. Also determine the values of a, if any, fortend to zero as twhich all (nonzero) solutions become unbounded as tThere is no value of a for which all solutions will tend to zeroas t → o∞.All solutions will tend to zero as t →∞ whenever:a Choose one vThere is no value of a for which all solutions will becomeO unbounded as tAll (nonzero) solutions will become unbounded as t → ∞ whenever:a Choose one v

Given: Let us consider the given second-order homogenous equation yιι-2α-14yι+α2-14α+40y=0.

Answered: Determine the values of a, if any, for…

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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Match the solution curve with one of the differential equations. O y" + 2y' + 2y = 0 O y" – 7y' + 12y = 0 О у" + 2y' +у %3D 0 О у" - 4y' - 5у %3D0 O y" + y = 0 O y" + 4y = 0 Explain your reasoning. (Assume that k, k,, and k, are all positive.) O The auxiliary equation should have one positive and one negative root, so that the solution has the form. + The differential equation should have the form y" + k?y = 0 where k = 1, so that the period of the solution is 2n. O The auxiliary equation should have two positive roots, so that the solution has the form c,ekı* + c,ek2*. O The auxiliary equation should have a repeated negative root, so that the solution has the form + O The differential equation should have the form y" + k?y = 0 where k = 2, so that the period of the solution is T. O The auxiliary equation should have a pair of complex roots a ± ßi where a < 0, so that the solution has the form eax(c, cos(Bx) + c, sin(Bx)).
3.) Given that y, (t) = (t + 2)- e' and y,(t) = e' - 2 are both solutions of a certain second order homogeneous linear differential equation: y" + Ay' + By = 0 Answer each question below. For each true/false question, state a brief reason that justifies your answer. a.) Show that these two solutions are linearly independent. b.) True or False: y, and y, form a fundamental set of solutions of this equation. c.) True or False: y,(t) = (t + 3) · e – 2 is also a solution of the equation d.) True or False: y,(e) = (t + 1) - e' is also a solution of the equation
Match the solution curve with one of the differential equations. y" + 4y = 0 О у" + 2y' + у %3D 0 O y" + y = 0 O y" – 2y' – 3y = 0 О у" - 7y' + 12у %3D 0 y" + 2y' + 2y = 0 Explain your reasoning. (Assume that k, k1, and k2 are all positive.) O The auxiliary equation should have a repeated negative root, so that the solution has the form cje-kx + c2xe¯kx. The auxiliary equation should have a pair of complex roots a ± ßi where a < 0, so that the solution has the form e"x(c1 cos Bx + c2 sin Bx). The differential equation should have the form y" + k?y = 0 where k = 1, so that the period of the solution is 2n. O The auxiliary equation should have one positive and one negative root, so that the solution has the form cqek1× + c2e¬k2×, The differential equation should have the form y" + k2y = 0 where k = 2, so that the period of the solution is n. O The auxiliary equation should have two positive roots, so that the solution has the form c1eki× + c2ek2×.
Use the concept that y = c, -∞0 < x < ∞, is a constant function if and only if y' = 0 to determine whether the given differential equation possesses constant solutions. (Enter all constant solutions below. If there are no constant solutions, enter NONE.) 3xy' + 2y = 4 y =
If f: X→ Y and B, C e P (Y), then -f-" (B) f1 (~ B) and f-1 (B - C) = f1 (B) ~f-1 (C)
One solution of the differential equation -y" +7y'- 10y = 0 5e2". is y = e. A second linearly independent solution is y Select one: O True False
Find three linearly independent solutions of the given third-order differential equation and write a general solution as an arbitrary linear combination of them. z""+11z" +24z" - 36z=0 A general solution is z(t) = B
Which of the following differential equation has a solution as y = cje2z + Cze3z + C3re3z ? O a. y" + 8y" + 21y' – 18y = 0 а. O b. y" – 8y" – 21y' – 18y = 0 %3D O c. y" + 8y" + 21y' + 18y = 0 O d. y" – 8y" + 21y' + 18y = 0 %3D O e. y" – 8y" + 21y' – 18y = 0
Find three linearly independent solutions of the given third-order differential equation and write a general solution as an arbitrary linear combination of them. z' + 3z" - 9z' – 27z = 0 A general solution is z(t) =
Determine the two singular points of the differential equation (x² - 81)y" + (9 − x)y′+ (x² + 18x +81)y = 0 List the points in increasing order: x1 = x2= Which of the following statements correctly describes the behaviour of the solutions of the differential equation near the singular point x1: A. All non-zero solutions are unbounded near 1. B. All solutions remain bounded near x1. C. At least one non-zero solution remains bounded near x1 and at least one solution is unbounded near x1. Which of the following statements correctly describes the behaviour of the solutions of the differential equation near the singular point 2: O A. All solutions remain bounded near 2. B. At least one non-zero solution remains bounded near 2 and at least one solution is unbounded near x2. C. All non-zero solutions are unbounded near 2.
Find three linearly independent solutions of the given third-order differential equation and write a general solution as an arbitrary linear combination of them. y""'-4y" -9y' + 36y=0 A general solution is y(t) =

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