Question

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.

Answer #1

] 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 differential equation dy/ dt = f(y) and
suppose that y1 is a critical point, i.e., f(y1) = 0.
Show that the constant equilibrium solution y = y1 is
asymptotically stable if f 0 (y1) < 0 and unstable if f 0 (y1)
> 0.

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).

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).

Considering the differential equation dx/dt = y − x^2 , dy/dt =
y − x
What would be the Jacobian matrix J(x,y), as well as the
eigenvalues/types at each equilibrium.

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

If the equation dx/dt = f(x, y), dy/dt = g(x, y) has a locally
stable equilibrium at the origin (0, 0), does the Jacobian matrix
J(x, y) satisfy: det J(0, 0) > 0, Tr J(0, 0) < 0 and why?

DIFFERENTIAL EQUATIONS PROBLEM
Consider a population model that is a generalization of the
exponential model for population growth (dy/dt = ky). In the new
model, the constant growth rate k is replaced by a growth rate r(1
- y/K). Note that the growth rate decreases linearly as the
population increases. We then obtain the logistic growth model for
population growth given by dy/dt = r(1-y/K)y. Here K is the max
sustainable size of the population and is called the carrying...

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'= 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.

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