Consider the vector field $F(x,y,z) = (-6y, -6x, z)$.
Show that $\mathbf{r}(t) = (e^{-6t} + e^{6t}, e^{-6t} - e^{6t}, e^{t})$ is a flowline for the vector field $F$.
That is, verify that (I trust you here)
$\mathbf{r'}(t) = F(\mathbf{r}(t)) = ($ , , $)$.

Now consider the curve $\mathbf{r}(t) = (\cos(-6t), \sin(-6t), e^{t})$. It is not a flowline of the vector field $F$, but of a vector field $G$ which differs in definition from $F$ only slightly.
$G(x,y,z) = ($ , , $)$.