Creating a path with differential equations I'm currently trying to create a 3-D vector field based on an equation. Once I create a vector field, how to actually make the object move along the vector field when placed at a point? 

My Approach:
Every vector field has an associated system of differential equations. For example, for a vector field, $$\vec A = x \hat{i}+y \hat{j}+z \hat{k}$$ the associated system differential equations are, $$\frac{dx}{dt}=x \\ \frac{dy}{dt} = y\\ \frac{dz}{dt}=z$$ and the corresponding solutions are $$x(t)=x_0 e^t\\ y(t)=y_0e^t \\ z(t)=z_0e^t$$

So now, we have path(i.e.) by varying t, we obtain a set of $(x,y,z)$.

Is there a way to solve this system of ODEs with blender. I found an answer, that was relevant, Can you animate objects using differential equations?, but I couldn't understand how to implement all those in this context.
When I have understood you correctly, you have already the solution of your diff eq, and you know x0, y0, z0 as parameters of this solution, defining the start location. When you run under Python a loop by incrementing t in discrete steps, you will be able to create a 3d chain of vertices with these coordinates and use this polygon as path for animation (convert to a curve, if needed). Since I am a newbie myself, I cannot provide any code, however.

Greetings from Germany, Tschöbbel