## Finding flow lines for a given vector field

Thought I'd ask here as I can't seem to find much in the way of online resources.

Suppose I'm given some arbitrary vector field $F(x,y)$ or $F(x,y,z)$. How would one go about (analytically) finding the family of flow lines for such a field? I know that a path $c(t)$ is a flowline of some field $F$ if

$\frac{dc}{dt} = F[c(t)]$

So I'm assuming the problem reduces to solving a separable ODE, but I can't say I'm terribly familiar with solving such DE's with vector functions.

Any suggestions?