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

[exmachina]

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)?

 

[n1person]

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.

 

[matematikawan]

If you want a numerical method to solve it, there already a similar problem. But you need to supply initial conditions.

https://www.physicsforums.com/showthread.php?t=345085

 

[gato_]

Are you sure about the signs? this is unstable anywhere. Local analysis can help you. If you set the system
$$x_{1}\equiv x, x_{2}\equiv x',x_{3}\equiv y,x_{4}\equiv y'$$
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))