"Since integral curves can’t cross an integral curve itself is both an upper and a lower fence"
Can anyone explain this for me a little more. I'm not sure if i'm getting it right.
I kind of understand this
"An integral curve is also a fence of both types"
The quote is a Remark in this http://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-i-first-order-differential-equations/geometric-methods/MIT18_03SCF11_s2_5text.pdf
MIT 18.03SC Differential Equations
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Did you get it?
An integral curve is a solution to a differential equation y' = f(x,y). Integral curves cannot cross each other because if they did cross each other, their slopes would be different at the point of intersection. And slopes of a solution cannot be different at any point, because then the differential equation is not a function (y' would be different for x and y of the point of intersection).
Since an integral curve cannot intersect another integral curve, it blocks other integral curves from crossing both from above and from below. Thus the document considers an integral curve as a "fence" of both types. Although this may be less useful than a fence of isoclines, because it is not easily obtainable from the differential equation.