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!