I understand that to solve flow lines for a gradient of single-variable components, one must parameterize it, apply separation of variables to each component differential equation, and solve for the constants using initial conditions. But, I'm stuck when trying to solve flow lines for gradients whose components have multiple variables.
For instance, consider the gradient $\langle xy, x^2y, y^2x\rangle$
In a write-up written at http://www.kkuniyuk.com/Math252FlowLines.pdf they took the gradient field $\langle x,y\rangle $, parameterized it as $\frac {dx}{dt} = x $ and $\frac {dy}{dt} = y $.
How can I parameterize it and apply separation of variables here if each component is dependent on multiple variables?
Edit: I was being dumb and overlooked that the process should still be the same.