Question

6) (8 pts, 4 pts each) State the order of each ODE, then
classify each of them as

linear/nonlinear, homogeneous/inhomogeneous, and
autonomous/nonautonomous.

A) Unforced Pendulum: θ′′ + γ θ′ + ω^2sin θ = 0

B) Simple RLC Circuit with a 9V Battery: Lq′′ + Rq′ +(1/c)q = 9

7) (8 pts) Find all critical points for the given DE, draw a phase
line for the system,

then state the stability of each critical point.

Logistic Equation: y′ = ry(1 − y/K), where r < 0

8) (6 pts) A mass of 2 kg is attached to the end of a spring and is
acted on by an

external, driving force of 8 sin(t) N. When in motion, it moves
through a medium that

imparts a viscous force of 4 N when the speed of the mass is 0.1
m/s. The spring

constant is given as 3 N/m, and this mass-spring system is set into
motion from its

equilibrium position with a downward initial velocity of 1 m/s.
Formulate the IVP

describing the motion of the mass. DO NOT SOLVE THE IVP.

9) (8 pts, 4 pts each) Find the maximal interval of existence, I,
for each IVP given.

A) (t^2 − 9) y′ − 7t^3 =√t, y(−2) = 12

B) sin(t) y′′ + ty′ − 18y = 1, y(4) = 9, y′(4) = −13

10) (30 pts, 10 pts each) Solve for the general solution to each of
the DEs given. Use an

appropriate method in each case.

A) Newton’s Law of Cooling: y′ = −k(y − T)

B) (sin(y) − y sin(t)) dt + (cos(t) + t cos(y) − y) dy = 0

C) ty′ − 5y = t^6 *e^t

Answer #1

1) State the main difference between an ODE and a PDE?
2) Name two of the three archetypal PDEs?
3) Write the equation used to compute the Wronskian for two
differentiable
functions, y1 and y2.
4) What can you conclude about two differentiable functions, y1 and
y2, if their
Wronskian is nonzero?
5) (2 pts) If two functions, y1 and y2, solve a 2nd order DE, what
does the Principle of
Superposition guarantee?
6) (8 pts, 4 pts each) State...

9) Find the maximal interval of existence, I, for each IVP
given.
A) (t^2 − 9) y′ − 7t^3 =√t, y(−2) = 12
B) sin(t) y′′ + ty′ − 18y = 1, y(4) = 9, y′(4) = −13

.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.

(25%) Problem 6: A mass m = 0.85 kg hangs at the end of a
vertical spring whose top end is fixed to the ceiling. The spring
has spring constant k = 85 N/m and negligible mass. The mass
undergoes simple harmonic motion when placed in vertical motion,
with its position given as a function of time by y(t) = A cos(ωt –
φ), with the positive y-axis pointing upward. At time t = 0 the
mass is observed to...

A mass of 4 Kg attached to a spring whose constant is 20 N / m
is in equilibrium position. From t = 0 an external force, f (t) =
et sin t, is applied to the system. Find the equation of motion if
the mass moves in a medium that offers a resistance numerically
equal to 8 times the instantaneous velocity. Draw the graph of the
equation of movement in the interval.

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...

3. A mass of 5 kg stretches a spring 10 cm. The mass is acted on
by an external force of 10 sin(t/2) N (newtons) and moves in a
medium that imparts a viscous force of 2 N when the speed of the
mass is 4 cm/s. If the mass is set in motion from its equilibrium
position with an initial velocity of 3 cm/s, formulate the initial
value problem describing the motion of the mass. Then (a) Find the...

When a mass of 4 kilograms is attached to a spring whose
constant is 64 N/m, it comes to rest in the equilibrium position.
Starting at t = 0, a force equal to f(t) = 80e−4t cos 4t is applied
to the system. Find the equation of motion in the absence of
damping.

A 1/4-kg mass is attached to a spring with stiffness 52 N/m.
The damping constant for the system is 6 N-sec/m. If the mass is
moved 3/4 m to the left of equilibrium and given an initial
rightward velocity of 1 m/sec, determine the
equation of motion of the mass y(t) =
and give
its damping factor,
quasiperiod, and
quasifrequency.

A mass m=0.65 kg hangs at the end of a vertical spring whose top
end is fixed to the ceiling. The spring has a constant K=85 N/m and
negligible mass. At time t=0 the mass is released from rest at a
distance d=0.35 m below its equilibrium height and undergoes simple
harmonic motion with its position given as a function of time
by y(t) = A cos(ωt – φ). The positive y-axis point
upward.
Part (b) Determine the value of the...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 3 minutes ago

asked 28 minutes ago

asked 32 minutes ago

asked 40 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago