Determine the field lines to the vector field $F(x, y) = (x^3 y, x y^3)$

Question

I can't find a proper way to determine the field lines to a vector field! For example, if I have the field $F(x, y) = (x^3 y, x y^3)$, how can I find its field lines? Any help would be much appreciated.

Answer 1

The field lines for a vector field $F$ are the set of lines that have $F$ as their tangent. These satisfy $$F_x dy=F_y dx$$

Now insert $F_x=x^3y$ and $F_y=xy^3$, separate and integrate. If you're still having troubles with it after this, or are seeking a proof of the relation I gave, let me know in the comments below.