Question

Consider the equation x'= x^{3} - 3x^{2} + 2x.
Sketch the phase line. Solve the equation and sketch the graphs of
some solutions including at least one solution with values in each
interval above, below and between the critical points. Identify the
critical points as stable or unstable.

Answer #1

Consider the equation:
x'=x^3-3x^2+2x
sketch the phase line. solve the equation and sketch the graphs
of some solutions including at least one solution with values in
each interval above, below and between the critical points.
identify critical points as stable or unstable

For
the autonomous differential equation dy/dt=1-y^2, sketch a graph of
f(y) versus y, identify the equilibrium solutions identify them as
stable, semistable or unstable, draw the phase line and sketch
several graphs of solutions in the ty-plane.

] 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.

PLEASE PLEASE SHOW YOUR WORK
1a. Solve the equation: y = 3x2 -2x - 5
1b. Now use f(x) = x3 + 2x2 -5x - 6 to
list all of the potential rational zeros of this function AND find
the real zeros of f algebraically (show synthetic division at least
once) and use the to factor f.

Consider the autonomous first-order differential equation
dy/dx=4y-(y^3).
1. Classify each critical point as asymptotically stable,
unstable, or semi-stable. (DO NOT draw the phase portrait and DO
NOT sketch the solution curves)
2. Solve the Bernoulli differential equation dy/dx=4y-(y^3).

Let f(x) = x3 + 3x2 − 9x − 27 . The first
and second derivatives of f are given below.
f(x) = x3 + 3x2− 9x − 27 = (x − 3)(x +
3)2
f '(x) = 3x2 + 6x − 9 = 3(x − 1)(x + 3)
f ''(x) = 6x + 6 = 6(x + 1)
a.) Find the x-intercepts on the graph of f.
b.)Find the critical points of f.
c.) Identify the possible inflection points...

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...

Consider the function f(x) = x3 − 2x2 − 4x + 9 on the interval
[−1, 3].
Find f '(x). f '(x) = 3x2−4x−4
Find the critical values. x =
Evaluate the function at critical values. (x, y) =
(smaller x-value)
(x, y) =
(larger x-value)
Evaluate the function at the endpoints of the given
interval.
(x, y) =
(smaller x-value)
(x, y) =
(larger x-value)
Find the absolute maxima and minima for f(x) on the interval
[−1, 3].
absolute...

1. solve for x and graph solutions on a real number line
-7≤ 2x+5≤21
2. Solve for x and graph solution on a real number line
|7-3x|>2
3. write without absolute value and do not replace radical with
decimal representation:
|4- radical 17|
4. Solve for x and check
| 2x-11 | =31
Side note* | means absolute value for 4 its
absolute of 4- radical 17 and then absolute

Consider the equation below.
f(x) =
2x3 + 3x2
− 72x
(a) Find the interval on which f is increasing. (Enter
your answer in interval notation.)
Find the interval on which f is decreasing. (Enter your
answer in interval notation.)
(b) Find the local minimum and maximum values of f.
local minimum
local maximum
(c) Find the inflection point.
(x, y) =
Find the interval on which f is concave up. (Enter your
answer in interval notation.)
Find the...

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