One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with… - George Pólya

The differential equation of the first order
<math>\frac {dy}{dx} = f(x,y)</math>
...prescribes the slope <math>\frac {dy}{dx}</math> at each point of the plane (or at each point of a certain region of the plane we call the field"). ...a differential equation of the first order... can be conceived intuitively as a problem about the steady flow of a river: Being given the direction of the flow at each point, find the streamlines. ...It leaves open the choice between the two possible directions in the line of a given slope. Thus... we should say specifically "direction of an unoriented straight line" and not merely "direction."

In plausible reasoning the principal thing is to distinguish... a more reasonable guess from a less reasonable guess.

Analogy pervades all our thinking,