### Examine the behavior of the solution

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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.)

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