As we will study in the coming weeks, the one-dimensional motion of a mass attached to a spring without friction is governed by the linear second-order equation with constant coeffi- cients (1) d²x dt² +w²x = 0, where w² is a positive parameter. (a) Find the general solution to equation (1). You should have found a periodic solution: can you say what the period is? (b) Find the particular solution assuming that the mass is initially still in position to at t = 0, i.e., using the initial conditions x(0) = xo and d (0) = 0. (c) With friction, the motion is described by the equation d²x dt² dx + f +w²x = 0, dt where both f and w² are positive parameters. Find the general solution to equation (2) and sketch the behaviour of the solution in time (remember to discuss all the possible cases!). Is this a periodic function? If so, can you find the period?
As we will study in the coming weeks, the one-dimensional motion of a mass attached to a spring without friction is governed by the linear second-order equation with constant coeffi- cients (1) d²x dt² +w²x = 0, where w² is a positive parameter. (a) Find the general solution to equation (1). You should have found a periodic solution: can you say what the period is? (b) Find the particular solution assuming that the mass is initially still in position to at t = 0, i.e., using the initial conditions x(0) = xo and d (0) = 0. (c) With friction, the motion is described by the equation d²x dt² dx + f +w²x = 0, dt where both f and w² are positive parameters. Find the general solution to equation (2) and sketch the behaviour of the solution in time (remember to discuss all the possible cases!). Is this a periodic function? If so, can you find the period?
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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Transcribed Image Text:2. As we will study in the coming weeks, the one-dimensional motion of a mass attached to a
spring without friction is governed by the linear second-order equation with constant coeffi-
cients
(1)
d²x
dt²
+w²x=
= 0,
where w² is a positive parameter.
(a) Find the general solution to equation (1). You should have found a periodic solution:
can you say what the period is?
=
(b) Find the particular solution assuming that the mass is initially still in position to at
= 0, i.e., using the initial conditions x(0) = xo and d (0) = 0.
t =
(c) With friction, the motion is described by the equation
d²x
dt²
dx
dt
+ f
+w²x = 0,
(2)
where both f and w² are positive parameters. Find the general solution to equation (2) and
sketch the behaviour of the solution in time (remember to discuss all the possible cases!). Is
this a periodic function? If so, can you find the period?
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