Question

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

Answer #1

Consider the differential equation u ′′ + bu = sin^2 (2t).
(a) For what value of b does the solution show resonance?
Clearly, explain your answer. Hint: sin2 x = 1−cos(2x) 2
(b) Consider the same differential equation but take the
non-homogeneous term to be sin4 t. For what value(s) of b does the
solution show resonance? You do not have to give an exact solution
but you do have to explain clearly your thinking using the
structure of the...

A function y(t) satisfies the differential
equation
dy
dt
= y4 − 11y3 + 24y2.
(a) What are the constant solutions of the equation? (Enter your
answers as a comma-separated list.)
y =
(b) For what values of y is y increasing? (Enter
your answer in interval notation.)
y
(c) For what values of y is y decreasing? (Enter
your answer in interval notation.)
y

Exercise 9.
In the questions below you can describe the relations/functions
either by drawing a diagram, by a formula, or by listing the
ordered pairs. Explain your solutions.
(i) Give an example of two sets A and B and a relation R from A to
B which is not a function.
(ii) [hard] Find a set A, |A| = 4 and deﬁne a bijective function
between A and P(A)? If such a set doesn’t exist give a reason.
Exercise 11....

Consider the nonlinear second-order differential
equation 4x"+4x'+2(k^2)(x^2)− 1/2 =0, where k > 0 is a constant.
Answer to the following questions.
(a) Show that there is no periodic solution in a simply connected
region R={(x,y) ∈ R2 | x <0}. (Hint: Use the corollary to
Theorem 11.5.1>>
If symply connected region R either contains no critical points of
plane autonomous system or contains a single saddle point, then
there are no periodic solutions. )
(b) Derive a plane autonomous system...

Instructions: Answer the questions below. Press
[Shift + Enter] after each question to answer.
What is a function in programming?
=>
How does a function return a value?
=>
Parameters are an important concept in defining
functions.
(a) What is the purpose of parameters? =>
(b) What is the difference between a formal parameter and an actual
parameter(argument)? =>
(c) In what ways are parameters similar to and different from
ordinary variables? =>
4 Discussion: In your own words, describe...

A. Answer the theoretical questions below
a. Explain the different types of market structures using both
graphs and verbal explanation. Provide a specific example of each
market.
b. Why monopolies arise and what governments can do in face of
monopolies?
c. Explain briefly how the price floor works in a market if the
government applies it. Use a specific real life example to argue
for the application of it.

Use the following linear regression equation to answer the
questions.
x3 = −17.3 + 3.7x1 +
9.6x4 − 1.7x7
(e) Suppose that n = 20 data points were used to
construct the given regression equation and that the standard error
for the coefficient of x4 is 0.820. Construct a
90% confidence interval for the coefficient of
x4. (Round your answers to two decimal
places.)
lower limit
upper limit
(f) Using the information of part (e) and level of significance 5%,...

Use the following linear regression equation to answer the
questions.
x3 = −16.9 +
3.9x1 +
9.8x4 −
1.9x7
(e) Suppose that n = 19 data points were used to
construct the given regression equation and that the standard error
for the coefficient of x4 is 0.883. Construct a
90% confidence interval for the coefficient of
x4. (Round your answers to two decimal
places.)
lower
limit
?
upper limit
?
(f) Using the information of part (e) and level of...

4)Physically, damping is always present. Now
equation 2 below is modified for damping.
x''+2x'+16x=3sin(t)
(2),
Solve in Mathematica or Matlab and plot the solutions for the
following conditions.
Using different styles and line color. Plot (a) and (b) on the
same graph then (c) and (d) on another.
= 8 x(0) = x’(0) = 0
= 4 x(0) = x’(0) = 0
= 4 x(0) = 0, x’(0) = 5
= 4 x(0) = 1, x’(0) = 0
To...

Important Instructions: (1) λ is typed as lambda. (2) Use
hyperbolic trig functions cosh(x) and sinh(x) instead of ex and
e−x. (3) Write the functions alphabetically, so that if the
solutions involve cos and sin, your answer would be
Acos(x)+Bsin(x). (4) For polynomials use arbitrary constants in
alphabetical order starting with highest power of x, for example,
Ax2+Bx. (5) Write differential equations with leading term
positive, so X′′−2X=0 rather than −X′′+2X=0. (6) Finally you need
to simplify arbitrary constants. For...

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