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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

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
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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urgently need part a and B

Question 3 Recall the difterentiation property for r(t) and its Laplace transform X(s): dr(t) + sX(s) – r(0-) (1) dt d" r(t) + s"X(s) -Es"-k lk-1)(0-) (2) dt" k=1 Use 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) d and D2: dt d In this expression D dt2 Consider a mass m that is hanging on a spring from a ceiling. The forces acting on the mass are gravity mg, the restoration of the spring k(d+y), and an external force r(t). Here d is the distance that the mass hangs from the natural length of the spring (i.e, it is the equilibrium distance for the ceiling) 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 as dy(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 should assume that r(t) is a causal signal, and remember that the transfer function focuses on the setting of zero initial conditions). dy? d) What is the zero-input response Y (s) of the system for y(0) = yo and dt2 (0) = 0?
Transcribed Image Text:Question 3 Recall the difterentiation property for r(t) and its Laplace transform X(s): dr(t) + sX(s) – r(0-) (1) dt d" r(t) + s"X(s) -Es"-k lk-1)(0-) (2) dt" k=1 Use 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) d and D2: dt d In this expression D dt2 Consider a mass m that is hanging on a spring from a ceiling. The forces acting on the mass are gravity mg, the restoration of the spring k(d+y), and an external force r(t). Here d is the distance that the mass hangs from the natural length of the spring (i.e, it is the equilibrium distance for the ceiling) 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 as dy(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 should assume that r(t) is a causal signal, and remember that the transfer function focuses on the setting of zero initial conditions). dy? d) What is the zero-input response Y (s) of the system for y(0) = yo and dt2 (0) = 0?
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