Question

DIFFERENTIAL EQUATIONS

1. A force of 400 newtons stretches a spring 2 meters. A mass of
50 kilograms is attached to the end of the

spring and is initially released from the equilibrium position with
an upward velocity of 10 m/s. Find the equation of

motion.

2. A 4-foot spring measures 8 feet long after a mass weighing 8
pounds is attached to it. The medium through

which the mass moves offers a damping force numerically equal to
times the instantaneous velocity. Find the

equation of motion if a mass of ¼ slug is initially released from
the equilibrium position with a downward

velocity of 5 ft/s.

3. A mass weighing 16 pounds stretches a spring feet. The mass
is initially released from rest from a point 2 feet

below the equilibrium position, and the subsequent motion takes
place in a medium that offers a damping force

that is numerically equal to 1/2 the instantaneous velocity.

The mass is driven by an external force equal to F(t) = 10 cos
3t

(a) The complementary solution is:
_________________________________________

(b) The particular solution is:
_______________________________________________

(c) The general solution is:
___________________________________________________

4. A beam is free on left and embedded on right, and
*w**x**=**w**0**,
0<x<L*, and governed by
*EI**d**4**y**d**x**4**=w**x*

Find y(x).

Hint: use boundary conditions in this order:

Boundary condition:

Boundary condition:

Boundary condition:

Boundary condition:

5. Find the first 4 terms of the two power series solutions of the DE about the ordinary point x = 0.

y''-2xy'+y=0

You can write the coefficients of your two answers as numbers
rather than in terms of Co and C1

That is y(x) = 1- 1/4x^2 + ..... instead of y(x) = Co [ 1- 1/4x^2 + ....]

6. Find the first 3 terms of the two power series solutions to the IVP about the ordinary point x = 0.

You can write the coefficients of your two answers as numbers rather than in terms of

(x-1)y'' -xy' + y = 0 , y(0) = -2, y' (0) = 6

7. Find the indicial roots of 3.

Roots: ___, _____

What do we expect the two series solutions to look like? Fill in
the blanks.

y = ____ * ( Co + C1x + C2x + ........ ) + _______*(bo + b1x +b2x +.....)

Answer #1

A force of 400N stretches a string 2 meters. A mass of 50kg is
attached to the end of the spring and stretches the spring to a
length of 4 meters. The medium the spring passes through creates a
damping force numerically equal to half the instantaneous velocity.
If the spring is initially released from equilibrium with upward
velocity of 10 m/s.
a) Find the equation of motion.
b) Find the time at which the mass attains its
extreme displacement...

A force of 64 pounds stretches a spring 4 feet. A mass of 4
slugs is attached to the spring and is initially released from rest
2 feet below the equilibrium position. (a) Suppose the spring has a
damping force equal to 16 times the instantaneous velocity and is
being driven by an external force, ?(?) = 4 cos(5?) . Write the IVP
that this problem describes. (3 pts) (b) Solve the equation in part
(a) to obtain the equation...

A mass weighing 16 pounds stretches a spring
8
3
feet.
The mass is initially released from rest from a point
3 feet
below the equilibrium position, and the subsequent motion takes
place in a medium that offers a damping force that is numerically
equal to
1
2
the
instantaneous velocity. Find the equation of motion
x(t)
if
the mass is driven by an external force equal to
f(t) = 20 cos(3t).
(Use
g =
32 ft/s2
for
the acceleration...

A mass weighing 16 pounds stretches a spring 8 3 feet. The mass
is initially released from rest from a point 3 feet below the
equilibrium position, and the subsequent motion takes place in a
medium that offers a damping force that is numerically equal to 1 2
the instantaneous velocity. Find the equation of motion x(t) if the
mass is driven by an external force equal to f(t) = 10 cos(3t).
(Use g = 32 ft/s2 for the acceleration...

A mass weighing 16 pounds stretches a spring 8/3 feet. The mass
is initially released from rest from a point 3 feet below the
equilibrium position, and the subsequent motion takes place in a
medium that offers a damping force that is numerically equal to 1/2
the instantaneous velocity. Find the equation of motion x(t) if the
mass is driven by an external force equal to f(t) = 10 cos(3t).
(Use g = 32 ft/s^2 for the acceleration due to...

A mass weighing 19.6 N stretches a spring 9.8 cm. The
mass is initially released from
a point 2/3 meter above the equilibrium position with a downward
velocity of 5
m/sec.
(a) Find the equation of motion.
(b)Assume that the entire spring-mass system is submerged in a
liquid that
imparts a damping force numerically equal to β (β > 0) times the
instantaneous
velocity.
Determine the value of β so that the subsequent motion is
overdamped.

A mass of 1 slug, when attached to a spring, stretches it 2 feet
and then comes to rest in the equilibrium position. Starting at t =
0, an external force equal to f(t) = 4 sin(4t) is applied to the
system. Find the equation of motion if the surrounding medium
offers a damping force that is numerically equal to 8 times the
instantaneous velocity. (Use g = 32 ft/s2 for the
acceleration due to gravity.)
What is x(t) ?...

A force of 720 newtons stretches a spring 4 meters. A mass of 5
kg is attached to the end of the spring and is released from a
position .5 meters below the equilibrium position with a downward
velocity of 8 meters per second. What is the equation of
motion?
Please solve using Differential equations

Determine C1 and C2 of the following damped motion
A 4-lb weight stretches a spring 4 ft. Initially the weight
released from 2ft above equilibrium position with downward velocity
2 ft/sec. Find the equation of motion x(t), provided that the
subsequent motion takes place in a medium that offers a damping
force numerically equal to (1/2) times the instantaneous
velocity

A mass weighing 32 pounds stretches a spring 2 feet. The mass is
initially released from rest from a point 1 foot below the
equilibrium position with an upward velocity of 2ft/sec. find the
equation of the motion and solve it, determine the period and
amplitude.

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