i(t) = y – x³ ý(t) = -x³ – y3 a) Is the origin stable or asymptotically stable for the linearized system? Can we use this to conclude the same for the original nonlinear system? Why or why not? b) Show that the “energy function" E(x, y) = 5 + is non-increasing along the solution, i.e., using the multivariable chain rule verify that
i(t) = y – x³ ý(t) = -x³ – y3 a) Is the origin stable or asymptotically stable for the linearized system? Can we use this to conclude the same for the original nonlinear system? Why or why not? b) Show that the “energy function" E(x, y) = 5 + is non-increasing along the solution, i.e., using the multivariable chain rule verify that
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
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PART A
![Problem 3. Consider the differential equation
i(t) = y – x³
ý(t) = -x³ – y³
a) Is the origin stable or asymptotically stable for the linearized system? Can we use this to
conclude the same for the original nonlinear system? Why or why not?
b) Show that the "energy function" E(x, y) = 5
i.e., using the multivariable chain rule verify that
x4
+
4
is non-increasing along the solution,
d
E(x(t), y(t)) < 0
dt
c) Use part b) to show that the origin is stable, i.e., suppose that you are given R> 0, then
provide a sufficient condition on r > 0 such that, if x²(0) + y² (0) < r², then we can conclude
that the solution x(t), y(t) satisifes x2 (t) + y² (t) < R² for all t > 0. [Argue as done in class
for the simple harmonic oscillator system]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe25f92a2-1c4d-41a3-af19-0cdf00d27604%2F08b2666b-c922-4c96-9614-5a241b051289%2F9x64699_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 3. Consider the differential equation
i(t) = y – x³
ý(t) = -x³ – y³
a) Is the origin stable or asymptotically stable for the linearized system? Can we use this to
conclude the same for the original nonlinear system? Why or why not?
b) Show that the "energy function" E(x, y) = 5
i.e., using the multivariable chain rule verify that
x4
+
4
is non-increasing along the solution,
d
E(x(t), y(t)) < 0
dt
c) Use part b) to show that the origin is stable, i.e., suppose that you are given R> 0, then
provide a sufficient condition on r > 0 such that, if x²(0) + y² (0) < r², then we can conclude
that the solution x(t), y(t) satisifes x2 (t) + y² (t) < R² for all t > 0. [Argue as done in class
for the simple harmonic oscillator system]
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