Question

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.

Answer #1

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 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 differential equation y′′+ 9y′= 0.(
a) Let u=y′=dy/dt. Rewrite the differential equation as a
first-order differential equation in terms of the variables u.
Solve the first-order differential equation for u (using either
separation of variables or an integrating factor) and integrate u
to find y.
(b) Write out the auxiliary equation for the differential
equation and use the methods of Section 4.2/4.3 to find the general
solution.
(c) Find the solution to the initial value problem y′′+ 9y′=...

Choose the correct answers
If y1 and y2 are two
solutions of a nonhomogeneous equation ayjj+
byj+ cy =f (x), then
their difference is a solution of the equation
ayjj+ byj+ cy =
0.
If f (x) is continuous everywhere, then there
exists a unique solution to the following initial value
problem.
f (x)yj=
y, y(0) = 0
The differential equation yjj +
t2yj −
y = 3 is linear.
There is a solution to the ODE
yjj+3yj+y...

Consider the differential equation. Find the solution y(0) =
2.
dy/dt = 4t/2yt^2 + 2t^2 + y + 1

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?

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.

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 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...

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