Geometrical interpretation of this property

[brunob]

Hi there!

I have the following property:
If x(t) is a solution of \left\{ \begin{array}{l} \dot{x} = f(x) \\ x(t_0) = x_0 \end{array} \right. then the function y(t) = x(t+t_0) is a solution of the equation with initial data y(0) = x_0.

How could it be interpreted geometrically?
Thanks!

 

[Only a Mirage]

Geometrically, think of f as a vector field. Imagine it as a collection of arrows describing the velocity of a fluid; that is, for each x, f(x) is the velocity of the fluid at the point x.

Now, imagine a small pebble moving in this fluid. At each point x, the trajectory of the pebble must be tangent to the vector f(x) (why?). Dropping a pebble into the fluid at a specific point x_0 represents an initial condition of your ODE. The answer to your question is the following: it doesn't matter what time you drop the pebble at x_0; the pebble's resulting trajectory will always be the same, because the fluid velocity field is unchanging with time.

The image of the solution of y is just a copy of the image of x, but shifted in time -- just like the pebble's motion when dropped at time t_0 is identical to its motion when dropped at 0, but shifted in time.

 

[brunob]

Nice example! So, if the velocity is constant it means that the position x(t) is linear and so y(t) is, and due the way the functions are linked I can say that their graphics are parallels.
Let me know if I'm wrong.

Thanks!