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?