I need to sketch the phase image belonging to the following vector field (I'm sorry, I don't know the exact terms in English, so I have just freely translated them - thanks for sharing the correct terms, if you want to :))
$F(x,y) = \begin{pmatrix} \frac{1}{2x}\\ yx^2\\ \end{pmatrix} $
In order to draft the phase image, I have to know the flow $\phi(t, x, y)$ (again, freely translated) of the vector field. Thus, I solved the differential equations $\frac{d}{dt}x(t) = \frac{1}{2x}$ and $\frac{d}{dt} y(t) = yx^2$, which led me to the following result (using initial valus $y(0) = y_0$ and $x(0) = x_0$:
$\phi(t,x,y) = \begin{pmatrix} \sqrt{t + x^2}\\ ye^{tx^2}\end{pmatrix}$
My question is: How I am supposed to draft the phase image of this vector flow (it's dependending of three variables, after all) and is it the correct solutiona for the vector flow anyway?
Thanks for your help!