Solving Coupled ODEs: x(t) and y(t)

In summary, the conversation is discussing a coupled ODE with a non-linear system and a second derivative. There is no known analytical solution, but a numerical method can be used with initial conditions provided. The signs of the system may make it unstable and local analysis can be used to find critical points.
  • #1
exmachina
44
0
I have the following coupled ODE:

2x+y^2=d^2x/dt^2
2y+x^2=d^2y/dt^2

How would one solve for x(t), y(t)?
 
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  • #2
Although I am no expert, I don't think there is an analytical solution to this differential equation. It is a non-linear system, which makes it already really difficult. One could linearize it around (0,0), however I don't know how to deal with the fact it is a second derivative instead of a first... perhaps a more skilled person can help.
 
  • #4
Are you sure about the signs? this is unstable anywhere. Local analysis can help you. If you set the system
[tex]x_{1}\equiv x, x_{2}\equiv x',x_{3}\equiv y,x_{4}\equiv y'[/tex]
and then look for the critical points, where all four equations go to zero, and you linearize around those points, it turns out there is always an unstable direction, so that your simulations will crash too, unless you start exactly at the critical points ((0,0) and (-2,-2))
 

FAQ: Solving Coupled ODEs: x(t) and y(t)

What is the purpose of solving coupled ODEs for x(t) and y(t)?

Solving coupled ODEs for x(t) and y(t) allows us to model and understand the behavior of two variables that are interdependent and change over time. This can help us make predictions and analyze the relationship between these variables.

What are the steps involved in solving coupled ODEs for x(t) and y(t)?

The first step is to write the coupled ODEs in standard form. Then, we can use numerical methods such as Euler's method or Runge-Kutta methods to approximate the solutions. Finally, we can plot the solutions and analyze the behavior of x(t) and y(t).

Can coupled ODEs have multiple solutions for x(t) and y(t)?

Yes, coupled ODEs can have multiple solutions for x(t) and y(t) depending on the initial conditions and the parameters in the equations. These solutions can represent different behaviors and relationships between the variables.

How do you interpret the solutions of coupled ODEs for x(t) and y(t)?

The solutions of coupled ODEs can be interpreted as trajectories in a phase space, where the x-axis represents x(t) and the y-axis represents y(t). The shape and behavior of these trajectories can provide insights into the relationship between the variables.

What are some real-world applications of solving coupled ODEs for x(t) and y(t)?

Coupled ODEs can be used to model and study various phenomena in physics, chemistry, biology, economics, and many other fields. For example, they can be used to analyze predator-prey relationships, population dynamics, chemical reactions, and electrical circuits.

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