Question

1-Consider the following.

36y'' − y = 0,

y(−4) = 1, y'(−4) = −1

Find the solution of the given initial value problem.

y(t) = ?

2- Consider the vibrating system described by the initial value problem. (A computer algebra system is recommended.)

u'' + u = 9 cos ωt,

u(0) = 5, u'(0) = 4

3-A spring is stretched 6 in. by a mass that weighs 8 lb. The mass is attached to a dashpot mechanism that has a damping constant of

0.25 lb · s/ft and is acted on by an external force of 3 cos 2t
lb. (Use g = 32 ft/s^{2} for the acceleration due to
gravity. Let u(t),

measured positive downward, denote the displacement in feet of
the mass from its equilibrium position at time *t*
seconds.)

(a) Determine the steady-state response of this system.

u(t) =

(b) If the given mass is replaced by a mass *m*, determine
the value of *m* for which the amplitude of the steady state
response is maximum.

m = slugs

Answer #1

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Differential Equations
A spring is stretched 6in by a mass that weighs 8 lb. The mass
is attached to a dashpot mechanism that has a damping constant of
0.25 lb· s/ft and is acted on by an external force of 2cos(2t)
lb.
(a) Find position u(t) of the mass at time t
(b) Determine the steady-state response of this system
Assume that g = 32 ft/s2

A spring is attached 6 in by a mass that weighs 8 lb. The mass
is attached to a dashpot mechanism that has a damping constant of
0.25 lb.s/ft and is acted on by an external force of 4cos(2t) lb.
1. Determine the steady state response of this system 2. If the
given mass is replaced by a mass m, determine the value of m for
which the amplitude of the steady state response is maximum. 3.
Write down the...

Consider a damped forced mass-spring system with m = 1, γ = 2,
and k = 26, under the influence of an external force F(t) = 82
cos(4t).
a) (8 points) Find the position u(t) of the mass at any time t,
if u(0) = 6 and u 0 (0) = 0.
b) (4 points) Find the transient solution uc(t) and the steady
state solution U(t). How would you characterize these two solutions
in terms of their behavior in time?...

Consider the initial value problem my′′+cy′+ky=F(t), y(0)=0,
y′(0)=0 modeling the motion of a spring-mass-dashpot system
initially at rest and subjected to an applied force F(t), where the
unit of force is the Newton (N). Assume that m=2 kilograms, c=8
kilograms per second, k=80 Newtons per meter, and F(t)=60cos(8t)
Newtons. Solve the initial value problem. y(t)= help (formulas)
Determine the long-term behavior of the system. Is limt→∞y(t)=0? If
it is, enter zero. If not, enter a function that approximates y(t)
for...

A spring with spring constant 4 N/m is attached to a 1kg mass
and a dashpot with damping constant 4 Ns/m.A periodic force equal
to 2 cos(t) N is applied to this system. Assume that the system
starts withx(0) = 1 andx′(0) = 2,

A mass weighing 32 lb is attached to a spring hanging from the
ceiling and comes to rest at its equilibrium position. At time t=0,
an external force of F(t) = 3cos(2t) lb is applied to the system.
If the spring constant is 10lb/ft and the damping constant is 4
lb-sec/ft, find the steady state solution for the system. Use g =
32 ft / sec^2

1 Consider an undamped mass-spring system with m = 1, and k =
26, under the influence of an external force F(t) = 82 cos(ωt).
a) (4 points) For what value of ω will resonance occur?
b) (6 points) In the case of resonance, solve the initial-value
problem with the system assumed initially at rest and briefly
discuss what happens.

Use the Laplace transform to solve the following initial value
problem,
y′′ − 5y′ − 36y =
δ(t − 8),y(0) = 0,
y′(0) = 0.
The solution is of the form
?[g(t)] h(t).
(a)
Enter the function g(t) into the answer box
below.
(b)
Enter the function h(t) into the answer box
below.

A mass weighing 96 lb is attached to a spring hanging from the
ceiling and comes to rest at its equilibrium position. At time
t=0, an external force of F(t) = 3cos(4t) lb is
applied to the system. If the spring constant is 10 lb/ft and the
damping constant is 3 lb-sec/ft, find the steady-state solution
for the system.
Use g=32 ft/sec^2

Consider the initial value problem
y' +
5
4
y = 1 −
t
5
, y(0) =
y0.
Find the value of
y0
for which the solution touches, but does not cross, the
t-axis. (A computer algebra system is recommended. Round
your answer to three decimal places.)
y0 =

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