Prove instability using Lyapunov function x' = x^3 + xy y' =

hadaasyj 2022-05-01 Answered
Prove instability using Lyapunov function
x=x3+xy
y=y+y2+xyx3
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Cristian Rosales
Answered 2022-05-02 Author has 26 answers
The dominant term is x6+x3y+y2. This term is positive (complete the square). All other terms are small when x,y, are small, so they can be controlled. Here are the details.
Write
V(x,y)=x6+x4y+y2y3xy2+x3y
=x6+(1+x)x3y+(1xy)y2
x6+(1+x)x3y+12y2if|x|+|y|14
x654|x|3|y|+12y2since|x|14
=(|x|358|y|)2+764y2
0
with equality iff x=y=0
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-01-02

Find dwdt using the appropriate Chain Rule. Evaluate dwdt at the given value of t. Function: w=xsiny, x=et, y=πt Value: t=0

asked 2022-04-08

Finding the extrema of a functional (calculus of variations)
I(y)=01y2y2+2xy dx 
Explanation:
Fy'=2y'  ddx(Fy')=2y''Fy=2y+2x
Which gives the Euler equation 2y +2y2x=0 (I think, unless I messed up my math).
Ok now I need to try to find solutions to this differential equation, however, I only know how to solve linear second order DE's. The x term is throwing me off and I am not sure how to solve this.
I know y(x)=x is a solution by inspection, but inspection is a poor man's approach to solving DE's.

asked 2022-04-29
y+16y=2sin(4x)
I try to solve this ode using the variation of parameters theorem.
The characteristic polynomial of the homogenous equation is r2+16=0.
Then u1(x)=sin(4x),u2(x)=cos(4x)
y(x)=c1(x)sin(4x)+c2(x)cos(4x)
I c1(x)sin(4x)+c2cos(4x)=0
II c1(x)cos(4x)c2sin(4x)=2sin(4x)
Multiply I by cos(4x) and II by sin(4x) and subtract.
c1(sin(4x)cos(4x)cos(4x)sin(4x))+c2(cos2(4x)+sin2(4x))=2
We get c2=2c2=2x,
c1=2cos(4x)sin(4x)=2cot(4x)c1=12ln|sin(4x)|.
I don't get why it incorrect, where am I wrong?
asked 2022-04-22
My textbook states we need locally Lipschitz continuous with respect to the second argument and uniformly with respect to the first argument of f(t, x). This is said to be equivalent to the result that for all compact subsets VU where URn+1 is open we have
L=(t,x)(t,y)VU|f(t,x)f(t,y)||xy|<
I understand that locally Lipschitz means there is a neighborhood around each point where f is Lipschitz, but what does the uniform part mean with respect to t?
asked 2022-04-23
Nonlinear ODE initial value problem
A student came to me with a problem I couldn't solve. It's the beginning of the semester in his Intro DiffEq class, and so the solution shouldn't be too difficult. But it completely stumped me, and now I can't let it go! Here it is:
Problem. Find all solutions to the IVP:
(dydx)24x2=x4y2,;;;;y(0)=0.
I thought about maybe just doing a sort of guess-and-check method, but the existence/uniqueness theorem doesn't apply, so I'm not sure that would help even if I could find a single solution, which I cannot in any case.
asked 2021-06-11
If x2+xy+y3=1 find the value of y''' at the point where x = 1
asked 2021-06-16

Use the substitution y=v to write each second-order equation as a system of two first-order differential equations (planar system). y+μ(t21)y+y=0