Vtyjkyuk

I need help finding the field lines of a vector field. I hesitant if the procedure and solution is correct.

The vector field is
$$mathbfF(x,y)=frac-yx^2+y^2mathbfhat x+fracxx^2+y^2mathbfhaty$$
So I should solve the equation
$$
mathbfF(mathbfr(t))=fracdmathbfr(t)dt, quad textwhere quad mathbfr(t)=x(t)mathbfhat x+y(t)mathbfhat y
$$
Therefore I have the equations$$
fracdx(t)dt=frac-yx^2+y^2 tag1
$$
$$
fracdy(t)dt=fracxx^2+y^2 tag2
$$
The first one is
$$
fracx^2+y^2-ydx=dt
$$
$$
Longrightarrow t=-y-fracx^33y+C_1 tag3
$$
And the second equation is
$$
fracx^2+y^2xdy=dt
$$
$$
Longrightarrow t=x+fracy^33x+C_2 tag4
$$
And let $(3)=(4)$ so
$$
-y-fracx^33y+C_1=
x+fracy^33x+C_2
$$
Let $-(C_1-C_2)=C_3$ so
$$
y+fracx^33y+x+fracy^33x+C_3=0
$$
Is this correct? How can i interpret this equation in the $(x,y,z)$-space?

From your (1) and (2) equation you can get:
$$
fracdydx = -fracxy
$$

I dont know how you obtained $(3)$ from $(1)$ and $(2)$. In any case $(3)$ and $(4)$ are false. The solution curves are in fact concentric circles, and you obtained third degree curves.

Note that your field is nothing else than $nablarm arg$, hence the field vectors are orthogonal to the level lines of $rm arg$, which are rays emanating from the origin.