What is the correct way to calculate streamlines of a vector field

Hello there,

What is wrong with my way of finding stream lines of a vector field? Say I have this vector field:

[itex]\vec{v} = x\,y\,\vec{i} + y\,\vec{j}[/itex]

You can see a plot here: http://kevinmehall.net/p/equationexp...0,10,-10,10%5D

It appears as if the stream lines could be [itex]y = log(x) + C[/itex].

I proceed to find out:

[itex]v_y \, \mathrm{d}x = v_x \, \mathrm{d}y\\
x\,y\,\mathrm{d}y = y\,\mathrm{d}x\\
\mathrm{d}y = \frac{1}{x}\,\mathrm{d}x\\
\int\,\mathrm{d}y = \int \frac{1}{x}\,\mathrm{d}x\\
y = log(x) + C
[/itex]

This looks about right. However there is a problem, when I look back at my vector field (http://kevinmehall.net/p/equationexp...0,10,-10,10%5D), for values of x less than zero, it appears as if the streamlines should be a mirror-image of y = log(x) + C.

So my question, does the above streamline calculation have more solutions which I have missed? Or is there something else which is wrong, which is causing me only to find the streamlines for x values greater than 0?

Thank you for your time.

Kind regards,
Marius