Solving Simple Coupled System Homework

In summary, the conversation discusses solving for f_{analytic} in a general system, as well as finding the eigenvalues, eigenmodes, and energies for this type of system. It suggests using a matrix form of the equations and diagonalizing the matrix to solve for the desired parameters.
  • #1
morenogabr
29
0

Homework Statement


given a general system,
[tex]
\frac{df}{dt}=k_{1}g(t)
[/tex]
[tex]
\frac{dg}{dt}=-k_{2}f(t)
[/tex]
How could one solve for [tex]f_{analytic}[/tex]. I've used wolfram, so I know what they look like. But how does one begin to solve for them?

Further, how does one find the eigenvalues, eigenmodes and energies for this type of system?

I have been assigned this problem from my research advisor and must admit I do not have much background with coupled systems.

 
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  • #2
Let
[tex]\mathbf{x} = \begin{pmatrix}f(t)\\g(t)\end{pmatrix}[/tex]
and express the system of equations in the form [itex]\mathbf{x}' = A\mathbf{x}[/itex], where A is a 2x2 matrix. That's the matrix you want to diagonalize.
 

FAQ: Solving Simple Coupled System Homework

1. What is a simple coupled system?

A simple coupled system is a set of equations that are interrelated and must be solved simultaneously to find a solution. This means that the variables in each equation are affected by the other equations in the system.

2. How do I solve a simple coupled system?

To solve a simple coupled system, you need to first identify the equations in the system and their variables. Then, you can use techniques such as substitution, elimination, or graphing to find a solution for the variables. It is important to carefully manipulate the equations and keep track of your steps to avoid errors.

3. What are the common mistakes made when solving simple coupled systems?

Common mistakes when solving simple coupled systems include forgetting to substitute correctly, making calculation errors, and not considering all possible solutions. It is important to double check your work and make sure that each step is accurate and makes sense in the context of the system.

4. Can a simple coupled system have more than two equations?

Yes, a simple coupled system can have any number of equations. The more equations in the system, the more complicated it may be to solve. However, the same techniques of substitution, elimination, or graphing can still be used to find a solution.

5. How can solving simple coupled systems be applied in real life?

Solving simple coupled systems can be applied in various fields such as physics, engineering, economics, and biology. For example, in physics, simple coupled systems can be used to model the motion of objects connected by springs. In economics, they can be used to analyze supply and demand relationships. In biology, they can be used to study the interactions between different species in an ecosystem.

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