Question

For the equationdydx=f(y) = (y−1)(y+ 2)2, do the following:(a) Draw the graph off(y) versusy.(b) Determine the values wherefhas zeros, and wheref′changes sign.(c) Draw the family of solutions to the differential equation.(c) Draw the phase line and classify the equilibrium solutions as stable,unstable, or semi-stable.(Do NOT solve this equation).

Answer #1

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).

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.

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

Consider the function f(x,y)=y+sin(x/y)
a) Find the equation of the tangent plane to the graph offat the
point(1,3)
b) Find the linearization of the function f at the point(1;3)and
use it to approximate f(0:9;3:1).
c) Explain why f is differentiable at the point(1;3)
d)Find the differential of f
e) If (x,y) changes from (1,3) to (0.9,3.1), compare the values
of ‘change in f’ and df

1.) Draw a neoclassical graph of a country in autarky
equilibrium. Label the graph carefully, and label the autarky
equilibrium point E. Suppose the international relative price of
the good on the Y-axis is lower than the country’s autarky relative
price (that is, (PY/PX)Aut >
(PY/PX)Int. Draw the international
price line on your graph. Label the new trade production point F,
and the new trade consumption point C. Which good will this country
import?

(a) Verify that all members of the family y =
(2)1/2 (c -
x2)-1/2 are solutions of the
differential equation.
(b) Find a solution of the initial-value problem.
y(x) is what

Consider the following nonlinear system: x'(t) = x - y, y'(t) =
(x^2-4)y
a. Determine the equilibria.
b. Classify the equilibria using linearization.
c. Use the nullclines to draw the phase portrait.
Please write neatly. Thanks!

Verify that all members of the family y =
(2)1/2 (c -
x2)-1/2 are solutions of the
differential equation
Find a solution of the initial-value problem. y' = (xy^3)/2 ,
y(0) = 9

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