Consider the fourth order ODE:Ly(4) – 256y = g(t)ly(0) = y(0) = y"(0) = y"(0) = 0.(i)the above ODE (2) (i.e., the Laplace transform of the solution L(y5,)(s) = ŷ5,(s) whenthe input g(t) = do(t)).Taking the Laplace transform, find the transfer function corresponding to(ii)form you found in part (i) (you may use the facts that L-1 )(t) = sinh(at) andL-()(t) = sin(at)).Find the impulse response of the ODE (2), yô, (t), by inverting the trans-а(iii)Now, write down the solution for a general g(t) in terms of y8, (t) and g(t).

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Consider the fourth order ODE: Sy(4) – 256y = 9(t) ly(0) = y'(0) = 3y"(0) = y"(0) = 0. | %3D (i) the above ODE (2) (i.e., the Laplace transform of the solution L(Y6o)(s) = ŷ§,(s) when the input g(t) = do(t)). Taking the Laplace transform, find the transfer function corresponding to %3D Find the impulse response of the ODE (2), yô, (t), by inverting the trans- (ii) form you found in part (i) (you may use the facts that L-1()(t) = sinh(at) and £1(3유교)(t) = sin(at)). %3D (iii) Now, write down the solution for a general g(t) in terms of y5, (t) and g(t).

Consider the fourth order ODE: Sy(4) – 256y = 9(t) ly(0) = y'(0) = 3y"(0) = y"(0) = 0. | %3D (i) the above ODE (2) (i.e., the Laplace transform of the solution L(Y6o)(s) = ŷ§,(s) when the input g(t) = do(t)). Taking the Laplace transform, find the transfer function corresponding to %3D Find the impulse response of the ODE (2), yô, (t), by inverting the trans- (ii) form you found in part (i) (you may use the facts that L-1()(t) = sinh(at) and £1(3유교)(t) = sin(at)). %3D (iii) Now, write down the solution for a general g(t) in terms of y5, (t) and g(t).

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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Consider the fourth order ODE:
Ly(4) – 256y = g(t)
ly(0) = y(0) = y"(0) = y"(0) = 0.
(i)
the above ODE (2) (i.e., the Laplace transform of the solution L(y5,)(s) = ŷ5,(s) when
the input g(t) = do(t)).
Taking the Laplace transform, find the transfer function corresponding to
(ii)
form you found in part (i) (you may use the facts that L-1 )(t) = sinh(at) and
L-()(t) = sin(at)).
Find the impulse response of the ODE (2), yô, (t), by inverting the trans-
а
(iii)
Now, write down the solution for a general g(t) in terms of y8, (t) and g(t).

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