Solving a nonlinear system with parameters

In summary, The conversation is about solving a system of nonlinear equations for an economics thesis on game theory. The speaker is unsure of how to solve the equations and is worried that it may be impossible with the given parameters, but notes that the system is symmetric. They mention the equations and ask for help in finding a solution. The expert suggests isolating y and substituting it into the second equation, which will result in a fourth degree polynomial for x.
  • #1
KevinL
37
0
I'm doing my economics thesis on game theory, and need to solve a system of nonlinear equations (these will be the reaction functions). Unfortunately, I have no idea how to solve it explicitly--indeed, I am a bit worried its impossible with the parameters. It is rather symmetric though.

The system is as such:

a0x2 + a1xy + a2x + a3y + a4 = 0

b0y2 + b1xy + b2x + b3y + b4 = 0

Any help pointing me in the right direction is much appreciated!
 
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  • #2
If all 'a' and 'b' constants are known, you can isolate, say, y from the first equation and insert the resulting into the second equation...a few line of expanding everything and multiplying, etc., you will end up with a forth degree polynomial for x
 

1. What is a nonlinear system with parameters?

A nonlinear system with parameters is a set of equations that involve variables raised to powers or multiplied together, along with constants known as parameters. These systems cannot be solved using traditional algebraic methods and require numerical methods.

2. How do you solve a nonlinear system with parameters?

To solve a nonlinear system with parameters, you can use numerical methods such as Newton's method, the bisection method, or the secant method. These methods involve iteratively guessing and refining the values of the parameters until a solution is found that satisfies all equations in the system.

3. What are the challenges of solving a nonlinear system with parameters?

One of the main challenges of solving a nonlinear system with parameters is that it can be difficult to determine a starting point for the numerical methods. The choice of initial values can greatly affect the convergence and accuracy of the solution. Additionally, these systems may have multiple solutions or no solutions at all.

4. Can a nonlinear system with parameters have an exact solution?

In most cases, a nonlinear system with parameters does not have an exact solution. This is because the equations involved are too complex to be solved algebraically. Instead, numerical methods are used to approximate the solution to a desired level of accuracy.

5. How are nonlinear systems with parameters used in scientific research?

Nonlinear systems with parameters are commonly used in scientific research to model real-world phenomena that cannot be described by simple linear equations. They are used in fields such as physics, engineering, economics, and biology to study complex systems and make predictions about their behavior.

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