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(d) u"(t)+u(t) = cos(t)

(d) u"(t)+u(t) = cos(t)

Advanced Engineering Mathematics
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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Do d,f,h
Exercise 4.3.2 For each ODE in parts (a)-(i): • Find a general solution un(t) to the homogeneous version of the ODE. ● Use the method of undetermined coefficients to find a particular solution up(t) to the nonhomogeneous ODE. However, in each of these the standard guess fails. Modify it appropriately, noting that the answer is not unique. • Write out a general solution to the nonhomogeneous ODE and use it to obtain the specific solution with initial conditions u(0) = 2 and u' (0) = 3. (a) u" (t) +9u' (t) +20u(t) = 2e-4t (b) 4u"(t)+24u' (t) +20u(t) = 8e¯¹ (c) 4u' (t) + 16u' (t) +12u(t) = 8e-³t (d) u" (t) +u(t) = cos(t) (e) u" (t) +2u' (t) +2u(t) = 2e-¹ sin(t) (f) u" (t) +2u' (t) + 10u(t)= e sin(3t) (g) u"(t) +4u' (t) +8u(t) = 16e-2 cos (21) (h) u"(t) +4u' (t) +4u(t)= te-21 (i) u" (t) +u(t) = sin(t) I Exercise 4.3.3 Consider the ODE mu" (t)+cu' (t)+ku(t)= eat and suppose that a is not a root of this ODE's characteristic equation. Show that a guess of the form up(t) = Ae for finding a particular solution will always work. Hint: iust substitute u.(t) into the ODE and show you can
Transcribed Image Text:Exercise 4.3.2 For each ODE in parts (a)-(i): • Find a general solution un(t) to the homogeneous version of the ODE. ● Use the method of undetermined coefficients to find a particular solution up(t) to the nonhomogeneous ODE. However, in each of these the standard guess fails. Modify it appropriately, noting that the answer is not unique. • Write out a general solution to the nonhomogeneous ODE and use it to obtain the specific solution with initial conditions u(0) = 2 and u' (0) = 3. (a) u" (t) +9u' (t) +20u(t) = 2e-4t (b) 4u"(t)+24u' (t) +20u(t) = 8e¯¹ (c) 4u' (t) + 16u' (t) +12u(t) = 8e-³t (d) u" (t) +u(t) = cos(t) (e) u" (t) +2u' (t) +2u(t) = 2e-¹ sin(t) (f) u" (t) +2u' (t) + 10u(t)= e sin(3t) (g) u"(t) +4u' (t) +8u(t) = 16e-2 cos (21) (h) u"(t) +4u' (t) +4u(t)= te-21 (i) u" (t) +u(t) = sin(t) I Exercise 4.3.3 Consider the ODE mu" (t)+cu' (t)+ku(t)= eat and suppose that a is not a root of this ODE's characteristic equation. Show that a guess of the form up(t) = Ae for finding a particular solution will always work. Hint: iust substitute u.(t) into the ODE and show you can
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