I have the question to sketch a phase portrait of the following, $$\dot{x} = 3y^2 - 25x^2$$ $$\dot{y} = 50xy$$

I have solved this to get $$y(y^2 - 25x^2) = C$$ To sketch the phase portrait I let C = 0 and so have a saddle point around (0,0) with the lines $x=0$, $y=5x$ and $y=-5x$.

I think this is okay, but then I am asked the following, in the case where $x(0) = -1$ and $y(0) = -6$ what is the value of $\frac{x(t)}{y(t)}$ in the limit as $t \rightarrow \infty$?

I have tried substituting in these values to my solution to get the constant = -66 but I am not sure where to go from there.

Thanks for any help.


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