Now do a nondimensional equation

We now choose a pressure driven velocity.  We can then divide out the pressure gradient.

How big is the second derivative term?

We see that for the nondimensionalized equation the (maximum) velocity is 0.5, the pressure gradient is -1.  Both of these have an order of magnitude of unity.

From "ueq2" above, the choice of the nondimensionalization of the pressure gradient tell us how changing the dimensional pressure gradient will change the dimensional velocity.  These are related linearly (with a minus sign on dpdx).  We did not have to solve anything to figure this out.    

In case you are interested, the average velocity is 1/3.