STEP rewards extracting behaviour before (or instead of) an explicit solution. From the right-hand side of dxdy=F(x,y) alone you can locate stationary points (where F=0), classify them, find limits as x→∞, and compare two equations by comparing their gradients, then sketch.
Trigger: the question says "sketch", "describe the behaviour as t→∞", or compares two similar ODEs.
Instances:
(i) set F=0 for stationary points and use the sign of F either side to call maxima/minima;
(ii) take x→∞ in the (often separable) solution to read off a horizontal asymptote or limiting value;
(iii) compare two integrands/gradients via a pointwise inequality (e.g. e−x3>e−x2 on 0<x<1) to place one solution curve above the other without solving either;
(iv) match the asymptotic form (e.g. y≈3−9/x) to fix the far-field shape of the sketch.
