### Examine the behavior of the solution

Assignment Help Mathematics
##### Reference no: EM132191767

Question 1. Determine if the zero solution of y' = (-1 + 1/(t+1))y is asymptotically stable. (Solve the equation and examine the behavior of the solution as t → ∞.) How does this compare with the behavior of the the zero solution of y' = -y?

Question 2. Consider the linear nonhomogeneous system Y' = AY +g(t). Assume that g(t) does not grow faster than an exponential function. That is, there exist real constants M > 0 , T ≥ 0 and an α such that |g(t)| ≤ Meαt for t > T. Then, show that every solution Φ(t) of the DE grows no faster than an exponential function. That is, show that there exits real constants K > 0 and β such that Φ(t)| ≤ Keβt for t ≥ T

Question 3. Discuss stability of the zero solution of y" + 2by' + ω2 sin y = 0 by studying the behavior of the zero solution of y" + 2by' + ω2y = 0. Cite the results from the text, that supports your conclusion.

Question 4. Consider the system of DE:

y'1 = -y1 - y2
y'2 = y1 - y23

Let V(y1, y2) = y12 + y22. Verify that V(y1, y2) is a Lyapunov function and with the help of a suitable theorem establish that the zero solution of the system of DE is asymptotically stable.

Question 5.

Exercise (a) In the (real or complex) system of differential equation

Y' = Ay + g(t, Y)

int A be a constant matrix and Re λ < α for every eigenvalue λ of A. Further, int g(t, y) be continuous for t ≥ 0, y ∈ Rn or Cn, and

|g(t, y)I ≤ h(t) |y|,

Where h(t) is a continuous (sufficient: locally integrable) function for t ≥ 0.

Show that every solution y(t) satisfies an estimate

|y(t)| ≤ K|y (0)|eαt+KH(t) with H(t) = 0t h(s)ds

for some constant K > 0 that is independent of y.

Hint. Derive an integral equation for Φ(t) = e-αt|y(t)| and use the generalized lemma of Gronwall.
From (a) conclude the following:

(b) If h(t) is integrable over 0 ≤ t < ∞ and if an eigenvalues of A have negative real part, then the solution y ≡ 0 is asymptotically stable and all solutions tend to zero as t → ∞.

(c) In the linear system
y' = (A + B(t))y,

let B(t) be a continuous matrix for t ≥ 0, and let

0 |B|(t)dt < ∞.

If all eigenvalues of A have negative real part, then the solution Y ≡ 0 is asymptotically stable.

Question 6. Discuss the stability of the zero solution of y' = αy + βy3 by considering a suitable linearized equation. (see page 316 of the text book.)

#### Main source of consumer enlightenment

Please provide a two to three sentence answer to each question. 1. According to the book, what can be argued as the glue that holds the supply chain together? 2. What is the m

#### What is their stated position regarding

What is their stated position regarding the issue? Do their actions reflect your position regarding the issue? Have they acted for the benefit of the 'common good' or do the

#### Trigonometric ratios

Trigonometric ratios, Learning the formula of trigonometric ratios creates confusion. There is trick to learn it i.e. “pandit bhadri Prasad har har bhole”  when we put it in f

#### Assignment on amazonfresh

Amazon has created three elements with how they are approaching the consumer market: fulfillment centers, Amazon Prime, and AmazonFresh. Analyze how Amazon has captured its

#### Amount of monthly loan payment

Sheila purchases a car for \$24,000, makes a down payment of 25%, and finances the rest witha 60-month car loan at an annual interest rate of 5.4% compounded monthly. What is

#### Find the expression the relates total capacitance

Capacitors can be used in sensing applications such as fuel gauges in airplanes and vibration sensors. The following is an application where a capacitive sensor can be used

#### What is the factor across the top

In order to divide polynomials, put your divisor on the left side and arrange your algebra tiles into a rectangular array so that the left edge matches your divisor. You may

#### How many students applied to both schools

In a group of 40 students, 25 applied to Columbia and 30 applied to Cornell. If 3 students applied to neither Columbia nor Cornell, how many students applied to both schools

### Write a Review 