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 critical points as stable or unstable

Answer #1

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Consider the equation x'= x3 - 3x2 + 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.

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.

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).

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

Solve the ODE: ( 3x^2 − 4xy − 3 ) + ( − 9y^2 − 2x^2 + 3 )y' = 0
.
Entry format: Write your solution equation so that: (1) The
equation is in implicit form. (2) The highest degree term
containing only x has a coefficient of 1. (3) Constants are
combined and moved to the RHS of the equation.
_____________________= C

3. Given the function ?(?) = (x^3/3)-(3x^2/2)+2x:
a. Find all critical numbers.
b. Identify which, if any, critical numbers are local max or min
and explain your answer.
c. Find any inflection points and give the x value.
d. On the interval [0.6, 2.6] identify the absolute max and min,
if any. and justify your answer.
e. Give the interval where the curve is concave up and justify
your answer.

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...

Given dy/dx =y(y−3)(1−y)^2 dx (a) Sketch the phase line
(portrait) and classify all of the critical (equilibrium) points.
Use arrows to indicated the flow on the phase line (away or towards
a critical point). (b) Next to your phase line, sketch the graph of
solutions satisfying the initial conditions: y(0)=0, y(0)=1,
y(0)=2, y(0)=3, y(2)=4, y(0)=5.

(b) Solve the separable differential
equation
y'=
(7e^-3x+2x^2-xcosx)/-6y^3
(c) Solve the IVP
x(1-siny)dy=(cosx-cosy-y)dx
.
y(π/2)=0

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