Let $F$ be the vector field given by
$$F (x, y, z) = \left(1 + x^2, −(1 + x^2)y, (1 + x^2)z\right).$$
How can I determine a parametrization for the field line of $F$, through $(0, 1, 1)\,$?
I was trying to solve 3 differential equations. But, for example, for the first one I get
$$x(t)= \tan(t+C(y,z)),$$
where $C(y,z)$ is a function that is constant in $x$, but that did not help me solve anything since I later need to integrate it.
You're trying to solve $$ \dot{x} = 1 + x^2$$ $$ \dot{y} = -(1+x^2)y$$ $$ \dot{z} = (1+x^2)z$$ The first equation, as you worked out, is separable and may be integrated. But this is a full differential equation for $x = x(t)$ (we're parametrising a field line by $t$), not a partial differential equation. So C is just an arbitrary constant.
You'll probably find it easier to integrate $$\dot{y} = -\dot{x}y$$ than using the solution for $x$. This gives $$\frac{dy}{dx} = -y$$ which is easily solved.