How do I set up difference equations for intersecting ovals?

In summary, the conversation is discussing a question about how many regions n simple ovals divide the plane into, given certain conditions. The participants are trying to determine the best approach to solving the problem, including using induction and setting up difference equations.
  • #1
morry
136
0
Ok, I am not sure if this is the right section, but seeing that difference equations are the discrete version...

Now the question is :
Into how many regions do n simple ovals divide the plane, given that every oval meets every other oval in two points and no point in the plane is common to more than two ovals.

I have NO idea how to go about this. Anyone care to shed some light?
 
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  • #2
Any ideas at all? Is this the right section?
 
  • #3
Are you sure this is the full question and doesn't have any other parts? What sort of work have you been doing recently on this topic? I'm just trying to get some context because the question doesn't seem very full.
 
  • #4
Looks to me like an induction question. Two ovals, meeting in two points, divide the plane into 4 regions. 3 ovals, each meeting each other in two points, divide the plane into 9 regions. Hmm, should we guess n2? I suspect you can find a difference equation relating the value for n ovals with the value for n+1 ovals.
 
  • #5
Thanks for the replies guys. :D

Halls, what would be some of the steps involved in setting up these types of equations? We have covered how to solve them, but not how to set them up. Which is what I am having trouble with.
 

1. What is a difference equation?

A difference equation is a mathematical equation that describes the relationship between the current value of a variable and its previous values. It is commonly used in fields such as engineering, economics, and physics to model systems that change over time.

2. How is a difference equation different from a differential equation?

While both difference equations and differential equations are used to describe changes in variables over time, they differ in how they model these changes. A difference equation uses discrete time steps, while a differential equation uses continuous time. In other words, a difference equation looks at how a variable changes from one specific point in time to the next, while a differential equation looks at how the variable changes at every point in time within a given interval.

3. What are the applications of difference equations?

Difference equations have many practical applications, including in engineering, economics, physics, biology, and social sciences. They can be used to model population growth, economic trends, chemical reactions, and many other real-world phenomena that involve changes over time.

4. How do I solve a difference equation?

The process for solving a difference equation depends on the specific equation and its variables. Generally, you will need to use algebraic manipulation and/or mathematical techniques such as difference operators, z-transforms, or Fourier transforms to find a solution. It is also common to use computer software, such as MATLAB or Mathematica, to solve more complex difference equations.

5. Can difference equations be used to predict future values?

Yes, difference equations can be used to predict future values of a variable based on its past values. However, the accuracy of the prediction will depend on the assumptions and parameters used in the equation, as well as the quality of the data used to create the equation. It is important to validate the results of a difference equation with real-world data before making any predictions or decisions based on the equation's output.

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