Question

A population ?(?) can be modeled by the differential equation ??/?? = 0.1? (1 − ?/400)

a. Find the equilibrium solutions and describe what each solution means for this population.

b. Describe the behavior of the function when ? > 400. Explain.

c. Describe the behavior of the function when 0 < ? < 400. Explain.

d. Sketch at least one solution illustrating each of parts (b) and (c) in the first quadrant below. Include the equilibrium solutions in your sketch.

Answer #1

Assuming that P≥0, a population is modeled by the differential
equation
dP/dt = 1.4P(1- P/3400)
1. For what values of P is the population increasing?
Answer (in interval notation):
2. For what values of P is the population decreasing?
Answer (in interval notation):
3. What are the equilibrium solutions? Answer (separate
by commas): P =

] Consider the autonomous differential equation y 0 = 10 + 3y −
y 2 . Sketch a graph of f(y) by hand and use it to draw a phase
line. Classify each equilibrium point as either unstable or
asymptotically stable. The equilibrium solutions divide the ty
plane into regions. Sketch at least one solution trajectory in each
region.

Consider the differential equation dy/dx= 2y(x+1)
a) sketch a slope field
b) Show that any point with initial condition x = –1 in the 2nd
quadrant creates a
relative minimum for its particular solution.
c)Find the particular solution y=f(x)) to the given differential
equation with
initial condition f(0) = 2
d)For the solution in part c), find lim x aproaches 0
f(x)-2/tan(x^2+2x)

Series Solutions of Ordinary Differential Equations For the
following problems solve the given differential equation by means
of a power series about the given point x0. Find the recurrence
relation; also find the first four terms in each of two linearly
independed sollutions (unless the series terminates sooner). If
possible, find the general term in each solution.
y"+k2x2y=0, x0=0,
k-constant

Use the differential equation u' = u(u - 4) to answer the
questions below
a)
Explain using the Picard Theorem that two graphs of solutions
for different differential equations do not intersect.
b)
Show that there exist exactly two different solutions that are
constant functions and find them
c)
You are given that u(0) = 1. Explain using the answers from a)
and b) that the solution is always a decreasing function

1. If x1(t) and x2(t) are solutions to the differential
equation
x" + bx' + cx = 0
is x = x1 + x2 + c for a constant c always a solution? Is the
function y= t(x1) a solution?
Show the works
2. Write sown a homogeneous second-order linear differential
equation where the system displays a decaying oscillation.

Consider the following second-order differential equation:
?"(?)−?′(?)−6?(?)=?(?)
(1) Let ?(?)=−12e^t. Find the general solution to the above
equation.
(2) Let ?(?)=−12.
a) Convert the above second-order differential equation into a
system of first-order differential equations.
b) For your system of first-order differential equations in part
a), find the characteristic equation, eigenvalues and their
associated eigenvectors.
c) Find the equilibrium for your system of first-order
differential equations. Draw a phase diagram to illustrate the
stability property of the equilibrium.

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...

Series Solution Method. Solve the given differential equation by
means of a power series about the given point x0. Find the
recurrence relation; also find the first four terms in each of two
linearly independent solutions (unless the series terminates
sooner). If possible, find the general term in each solution.
(1 − x)y′′ + y = 0, x0 = 0

Solve the given differential equation by means of a power series
about the given point x0. Find the recurrence relation; also find
the first four terms in each of two linearly independent solutions
(unless the series terminates sooner). If possible, find the
general term in each solution.
y′′ + xy = 0, x0 = 0

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