Question

Suppose the motion of a weight attached to a spring is given by the differential equation

y′′+ 6y+ 10y= 0

and

y(0) = 0, y′(0) = 3. Find the solution y(t) to this initial value problem.

Answer #1

. Write and solve a differential equation that models the motion
of a spring whose mass is 2a, spring constant b, and damping a,
where the numbers a is 3, b is 6. Assume that the initial position
is y = 1 and initial velocity is y 0 = −1. Write your solution as a
single, phase-shifted cosine function.

ﬁnd the general solution of the given differential equation
1. y''−2y'+2y=0
2. y''+6y'+13y=0
ﬁnd the solution of the given initial value problem
1. y''+4y=0, y(0) =0, y'(0) =1
2. y''−2y'+5y=0, y(π/2) =0, y'(π/2) =2
use the method of reduction of order to ﬁnd a second solution of
the given differential equation.
1. t^2 y''+3ty'+y=0, t > 0; y1(t) =t^−1

Find an equation of the curve that passes through the point
and has the given slope. (Enter your solution as an
equation.)
(0, 4), y' =
x
6y
2. Find the particular solution of the differential equation
that satisfies the initial condition. (Enter your solution as an
equation.)
Differential Equation Initial Condition
y(1 + x2)y' − x(7 + y2) = 0
y(0) =
3

MASS SPRING SYSTEMS problem (Differential Equations)
A mass weighing 6 pounds, attached to the end of a spring,
stretches it 6 inches.
If the weight is released from rest at a point 4 inches below
the equilibrium position, the system is immersed in a liquid that
offers a damping force numerically equal to 3 times the
instantaneous velocity, solve:
a. Deduce the differential equation that models the mass-spring
system.
b. Calculate the displacements of the mass ? (?) at all...

MASS SPRING SYSTEMS problem (Differential Equations)
A mass weighing 6 pounds, attached to the end of a spring,
stretches it 6 inches.
If the weight is released from rest at a point 4 inches below
the equilibrium position, and the entire system is immersed in a
liquid that imparts a damping force numerically equal to 3 times
the instantaneous velocity, solve:
a. Deduce the differential equation that models the mass-spring
system.
b. Calculate the displacements of the mass ? (?)...

.1.) Modelling using second order differential equations
a) Find the ODE that models of the motion of the dumped spring
mass system with mass m=1, damping coefficient c=3, and spring
constant k=25/4 under the influence of an external force F(t) = cos
(2t).
b) Find the solution of the initial value problem with x(0)=6,
x'(0)=0.
c) Sketch the graph of the long term displacement of the mass
m.

Consider the differential equation y′′+ 9y′= 0.(
a) Let u=y′=dy/dt. Rewrite the differential equation as a
first-order differential equation in terms of the variables u.
Solve the first-order differential equation for u (using either
separation of variables or an integrating factor) and integrate u
to find y.
(b) Write out the auxiliary equation for the differential
equation and use the methods of Section 4.2/4.3 to find the general
solution.
(c) Find the solution to the initial value problem y′′+ 9y′=...

A 128 lb weight is attached to a spring whereupon the spring is
stretched 2 ft and allowed to come to rest. The weight is set into
motion from rest by displacing the spring 6 in above its
equilibrium position and also by applying an external force F(t) =
8 sin 4t. Find the subsequent motion of the weight if the
surrounding medium offers a negligible resistance.

Logistic Equation The logistic differential equation y′=y(1−y)
appears often in problems such as population modeling.
(a) Graph the slope field of the differential equation between
y= 0 and y= 1. Does the slope depend on t?
(b) Suppose f is a solution to the initial value problem with
f(0) = 1/2. Using the slope field, what can we say about fast→∞?
What can we say about fast→−∞?
(c) Verify that f(t) =11 +e−tis a solution to the initial value
problem...

A 6lb wieght can stretch a spring 6 inches. Suppose the weight
is pulled 4 inches past the equilibrium point and released from
rest. The initial equation is y(t)=1/3*cos(8t)+0*sin(8t) Suppose
that a damping force given in pounds numerically by 1.5 times the
instantaneous velocity in feet per second acts on the 6lb weight.
Find the position x of the weight as a funtion of time.

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