Question

] 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.

Answer #1

The vertical axis is y and horizontal axis is t.

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 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).

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 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...

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).

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...

Consider the differential equation
y' =
y2 − 9
.
Let
f(x, y) =
y2 − 9
.
Find the partial derivative of f.
df
dy
=
Determine a region of the xy-plane for which the given
differential equation would have a unique solution whose graph
passes through a point
(x0, y0)
in the region.
A unique solution exits in the entire x y-plane.
A unique solution exists in the region −3 < y < 3.
A unique solution exits...

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 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.

differential equation(2x+3y)dx+ (y+2)dy=0

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