3. a) Prove that if y₁ is a solution to the differential equation ay" +by' + cy = fi(t) andy2 is a solution to ay" +by' + cy=f2(t), then for any constants k₁ and k2, the functionkıyı k2y2 is a solution to the differential equationay"+by+cy = k₁f1(t) + k₂f2(t).tb) Given that y₁ = (1/4) sin(2t) is a solution to y" + 2y' + 4y = cos(2t) and that y2 =is a solution to y" +2y' +4y = t, find the solution to y" +2y' +4y = 2t-3 cos (2t).(Use part a)48

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Answered: 3. a) Prove that if y₁ is a solution to…

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. VI Where Oy"+y=0 Oy" + 2y' + 2y = 0 Oy" - 5y' + 6y = 0 Oy" - 3y' - 4y = 0 Oy" + 16y = 0 Oy" + 2y' + y = 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 c₁e-kx + c2xe-kx O The auxiliary equation should have two positive roots, so that the solution has the form c₁ek1x + czek2x. O The differential equation should have the form y" + k²y = 0 where k = 1 so that the period of the solution is 2. O The differential equation should have the form y" + k²y = 0 where k = 2 so that the period of the solution is. O The auxiliary equation should have one positive and one negative root, so that the solution has the form c₁ek1x + c₂e-k2x. O The auxiliary equation should have a pair of complex roots a ± ßi where a < 0, so that the solution has the form exx (c₁ cos x + c2 sin BX).
2. The hatchery was recently bought out by a corporation and the new owners are planning on allowing people to fish at the hatchery. So, the previous differential equation will no longer be a good prediction of the fish population at a given time. Below, find three possible differential equations that model the population of the fish in the hatchery, under the new assumption that people will be allowed to fish at the hatchery. Which of the three differential equations best models the new situation and why did you choose this one? (Assume the constant k represents the annual harvesting rate.) (a) dP dr - 2P (1-P) dt 25 - kP (b) dP=2P (1-P₂-k) dt 25 (c) d=2P (1-P)-k dt 25 Based on the differential equation you chose from Question 2, prepare a one page report for the CEO of the corporation that explains the implications that various choices of the annual harvesting rate k will have on future fish populations. You may include graphical representations in your report, but you must explain,…
Verify that the indicated function y = p(x) is an explicit solution of the given first-order differential equation. (y-x)y=y-x+ 2; y=x+2√x+4 When y=x+2√x+4, y' = Thus, in terms of x, (y-x) = y-x+2= Since the left and right hand sides of the differential equation are equal when x + 2√x+4 is substituted for y. y=x+ 2√x+4 is a solution. Proceed as in Example 6, by considering o simply as a function and give its domain. (Enter your answer using interval notation.) Then by considering as a solution of the differential equation, give at least one interval I of definition, 0 (-8,4) O (-8,-4) 0 (-3,-4) ((-4, 0) 0 (-4,4) Need Help? Wash
Determine the ORDER, DEGREE and LINEARITY of the following differential equations.
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)).
Determine the values of a, if any, for which all solutions of the differential equation y"- (2a - 14)y + (a - 14a + 45)y = 0 | tend to zero as t o. Also determine the values of a, if any, for -> which all (nonzero) solutions become unbounded as t→ 00. There is no value of a for which all solutions will tend to zero as t → 0. All solutions will tend to zero as t-0∞ whenever: a Choose one There is no value of a for which all solutions will become unbounded as t o. All (nonzero) solutions will become unbounded as t → ∞ whenever: a Choose one
Solve the Linear Higher Order of differential equation using the formulas below. (xD — 2)(xD + 1)е-2х Formulas: 01. D"eаx — anеаx 02. Ф(D)eax — ф (а)еах %3D 03. Dª(eax)n % — еах (D + a)"и 04. Ф(D)(eаxи) %3D еахф (D + а)и 05. (D — а)"(ех и) еах D"u %3D
1 - Find the values of (h,k) so that the differential equation dy (4x+3y–15) (2а+ у -7) substitution x = X + h, y =Y+k. | can be solved by using the dx а) (3, 2) b)(2, 3) с) (3, 1) d)(1, 3)

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