urgently need part a and B Question 3Recall the difterentiation property for r(t) and its Laplace transform X(s):dr(t)+ sX(s) – r(0-)(1)dtd" r(t)+ s"X(s) -Es"-k lk-1)(0-)(2)dt"k=1Use this property to solve the following problems:a) Solve the following differential equation:(D² + 7D + 12)y(t) = (D+2)r(t)r(t) = 3e "u(t), y(0) =0, (0)=0(3)-2t(4)dand D2:dtdIn this expression Ddt2Consider a mass m that is hanging on a spring from a ceiling. The forces acting on the mass aregravity mg, the restoration of the spring k(d+y), and an external force r(t). Here d is the distancethat the mass hangs from the natural length of the spring (i.e, it is the equilibrium distance for theceiling) and y is the distance of the mass from this equilibrium. k is the spring constant.We can model the motion and position of the mass using the differential equation asdy(t)dt2-k(d+ y(t)) + mg +x(t).(5)This equation describes a system that relates the input force r(t) to an output position.b) How does the equilibrium position of the mass relate to m, g and k?c) What is the transfer function H(s) of the system relating y(t) and a(t)? (Here you shouldassume that r(t) is a causal signal, and remember that the transfer function focuses on thesetting of zero initial conditions).dy?d) What is the zero-input response Y (s) of the system for y(0) = yo anddt2(0) = 0?

Consider the given differential equation: and where

Answered: a) Solve the following differential…

a) Solve the following differential equation: (D² +7D + 12)y(t) = (D+2)r(t) r(t) = 3e-"u(t), y(0) = 0, ý(0) = 0 %3D In this expression D = and D2 = dt dt2

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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