Consider the differential equation dz dz m dt2 +p, + kz = F(t) dt which models an oscillator of mass m subject to a restoring force k, damping p and an external force F(t). The external force F(t) is given by F(t) = 18e-* and also m = 1, p = 4 and dz k = 13. Initial conditions are that z = 1 and = 4 when t = 0. dt (i) By taking the Laplace Transform of the differential equation show that 7, (the Laplace Transform of the solution) satisfies s2 + 10s + 34 (s + 2)(s² + 4s + 13) (ii) Hence find z(t). post (iii) Explain what happens to z(t) for large t.

Consider the differential equation dz dz m dt2 +p, + kz = F(t) dt which models an oscillator of mass m subject to a restoring force k, damping p and an external force F(t). The external force F(t) is given by F(t) = 18e-* and also m = 1, p = 4 and dz k = 13. Initial conditions are that z = 1 and = 4 when t = 0. dt (i) By taking the Laplace Transform of the differential equation show that 7, (the Laplace Transform of the solution) satisfies s2 + 10s + 34 (s + 2)(s² + 4s + 13) (ii) Hence find z(t). post (iii) Explain what happens to z(t) for large 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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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

Consider the differential equation
dz
dz
+p-
dt
+ kz = F(t)
m
dt2
which models an oscillator of mass m subject to a restoring force k, damping p and an external
force F(t). The external force F(t) is given by F(t) = 18e-* and also m = 1, p = 4 and
dz
k = 13. Initial conditions are that z = 1 and
= 4 when t = 0.
dt
(i) By taking the Laplace Transform of the differential equation show that 7, (the Laplace
Transform of the solution) satisfies
s? + 10s + 34
(s + 2)(s² + 4s + 13)
(ii) Hence find z(t).
post
(iii) Explain what happens to z(t) for large t.

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