ODE system question

Consider a system of the form: dx/dt = f(x,y) , dy/dt=g(x,y), with the property that the associated ODE dy/dx = g(x,y)/f(x,y) has a unique solution to IVP y(0)=0.

Also, f(x,y) is smooth every except the point (0,0), at which it has an infinite discontinuity, and g(x,y) is continuous everywhere. Does it follow that there is a solution to the system which tends to (0,0)?

This problem may be not well-formulated, but it seems like there may be a topological argument for the existence of such a solution to the system.

1 Answer

I may be missing some subtle point here, but it seems to me that if you let Y(x) be your presumed solution to Y'(x)=g(x,Y)/f(x,Y) and then let x(t) solve dx/dt=f(x,Y(x)) and put y(t)=Y(x(t)), you have your answer. The only way this could fail to tend to (0,0) for some value of t is if f(x,Y(x))=0 for arbitrarily small x, in which case I would question the validity of the assumed solution Y to begin with.

(Maybe I should have made this a comment rather than an answer, since the problem as I have understood it does not seem all that interesting. If you agree, feel more than free to not award any points to it, though I'd appreciate the absence of negative points.)