Finding Solutions for a Nonlinear System with Numerical Integration

Is the neighborhood of (0,0) the interval [0.01, 0.02]?To use Euler's method, you would start with some initial values for x and y, let's say x0 and y0. Then, you would use the differential equations to find x1 and y1, which would be the values of x and y at time t=1. Then, use those values to find x2 and y2, and so on. This will give you a numerical approximation of the solution to the system of equations.In terms of choosing initial values, you can choose them as close to (0,0) as you want, but they should not be exactly (0,0). For example,
  • #1
kreil
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Homework Statement


Given the system
[tex]x'(t)=-ax(t)+ky(t)+g[/tex]
[tex]y'(t)=lx(t)-by(t)+h[/tex]

If g=h=0,

a) Find the equilibrium
b) Show that if ab-lk does not equal 0, the steady state found in (a) is the only solution
c) choose a,b,l,k such that ab-lk > 0. Find numerically the solution of the system starting in a neighborhood of the equilibrium.


2. The attempt at a solution
a) If x'(t)=y'(t)=0, then ax(t)=ky(t) and lx(t)=by(t). This is true for ab = lk, i.e. ab-lk=0.

and then I run into trouble. I don't know how to explicity show (b), and have even less of an idea on how to start (c)

Help!
Josh
 
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  • #2
Should

"b) Show that if ab-lk does not equal 0, the steady state found in (a) is the only solution"

not read

"b) Show that if ab-lk does not equal 0, the steady state found in (a) is the only equilibrium solution"

Now, consider that an equilibrium or "steady state" solution correspond to x'=y'=0.
Finding such a solution here leads you to an algebraic linear system of equations.
Go back to and reuse the theory for linear systems of equations.
 
  • #3
I don't understand how I'm supposed to get values for x(t) and y(t).

Can you show me an example?
 
  • #4
You have, as you said in your first post, the two equations ax= ky and lx= by. Are you saying you do not know how to solve two linear, simultaneous equations?
 
  • #5
The part that confuses is me is that in (b) we prove that if ab-lk does not equal zero then x and y have only one equilibrium point..(0,0). Then part (c) says to solve the system for ab-lk>0, which would imply that (x,y)=(0,0).

Is this correct or am i missing something?
 
  • #6
One more time: can you solve the two equations ax= ky and lx= by for x and y? If you can, what happens to that solution if ab- lk DOES equal 0?

Yes, if ab-lk> 0 then (x,y)= (0,0) is the only EQUILIBRIUM solution. But you are asked to find a solution for an intial value in a neighorborhood of (0,0), not (0,0) itself.
 
  • #7
I don't think you understand my extreme inability to solve any part of this problem. Here is an attempt at (c):

a=0.3
b=0.2
l=0.1
k=0.4

Then ab-lk=0.02 > 0.

0.3 x(t) = 0.4 y(t)
0.1 x(t) = 0.2 y(t)

So, x(t) = 1.333 y(t)
y(t) = 0.5 x(t)

And I don't know what to do after this since it doesn't even seem to be true. Should I just plug in values of x(t),y(t) close to 0? I am not given any initial values. I really need help.
 
Last edited:
  • #8
kreil said:
I don't think you understand my extreme inability to solve any part of this problem.
I have asked you repeatedly to solve the linear equations ax= ky and lx= by. If you cannot do that then you should not be attempting differential equations!
From ax= ky, y= (a/k)x. Putting that into kx= by we have lx= (ab/k)x so that lkx= abx or (lk-ab)x= 0. If lk- ab is not 0 what is x equal to? What is y equal to? If lk- ab= 0 what can you say about x?

Here is an attempt at (c):

a=0.3
b=0.2
l=0.1
k=0.4

Then ab-lk=0.02 > 0.

0.3 x(t) = 0.4 y(t)
0.1 x(t) = 0.2 y(t)

So, x(t) = 1.333 y(t)
y(t) = 0.5 x(t)

And I don't know what to do after this since it doesn't even seem to be true. Should I just plug in values of x(t),y(t) close to 0? I am not given any initial values. I really need help.
Yes, it is true: x= (4/3)y and y= (1/2)x so x= (4/3)(1/2)X= (2/3)x. That is, (1/3)x= 0 so x= 0 and then y= 0. The only equilibrium solution is (0,0). Now choose a starting value for (x,y) that is "close to" (0,0) and numerically solve the system of equations.
 
  • #9
Thanks a lot for your help ivy, I have no other source to turn to on this problem. I have never done differential equations before and my teacher just gave us these projects without covering any of the material in class.

Did I solve the system correctly for these starting values? (I assume that I am to plug in one value in each equation and solve for the other variable.)

(x,y) = (0.01, 0.02)

0.3 * 0.01 = 0.4 * y(t) => y(t) = 0.0075

0.1 * x(t) = 0.5 * 0.02 => x(t) = 0.1

edit: Since I am no longer solving for the equilibrium, am I supposed to keep x' and y' in the system?
 
Last edited:
  • #10
Now choose a starting value for (x,y) that is "close to" (0,0) and numerically solve the system of equations.

This part is my biggest problem. I'm not exactly sure what constitutes a solution here. Am I to use Euler's method and approximate some values of x and y? Or am I trying to come up with a function that satisfies the diff eq? Any help at all here is appreciated!
 
  • #11
I really need help guys, any takers?
 
  • #12
Yes -- use Eulers method.
 
  • #13
When I choose values of x,y close to the equilibrium does t=0?
 

What is numerical integration?

Numerical integration is a method used in mathematics and scientific computing to approximate the value of a definite integral. It involves dividing the integral into smaller intervals and using numerical techniques to calculate the area under the curve.

Why is numerical integration important?

Numerical integration is important because it allows us to solve problems that cannot be solved analytically. It is also widely used in various fields such as physics, engineering, and economics to model and analyze real-world phenomena.

What are the different methods of numerical integration?

Some commonly used methods of numerical integration include the trapezoidal rule, Simpson's rule, and Gaussian quadrature. Each method has its own advantages and limitations, and the choice of method depends on the specific problem at hand.

How accurate is numerical integration?

The accuracy of numerical integration depends on various factors such as the method used, the number of intervals, and the smoothness of the function being integrated. In general, the more intervals used, the more accurate the approximation will be.

What are the applications of numerical integration?

Numerical integration has a wide range of applications in various fields such as physics, engineering, economics, and statistics. It is used to solve differential equations, evaluate complex integrals, and analyze data from experiments and simulations.

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