Direction field

To build it, choose a grid of points \((x,y)\), evaluate \(f(x,y)\) at each one and draw a short segment with that slope. Nothing needs to be solved: you only substitute. A solution is then any curve that, at each point it passes through, is tangent to the segment drawn there, like a leaf following the current of a river.

Curves along which the slope is constant are called isoclines. For \(y'=x+y\), the isocline of slope \(c\) is the line \(y=c-x\); drawing a few quickly organizes the field. Isoclines are not solutions: they only mark where the segments look the same. That one sometimes coincides with a solution, like \(y=x-1\) in \(y'=x-y\), is a coincidence of the example.

The field lets you read properties without formulas: where solutions increase or decrease, where there are constant solutions (points with \(f=0\) along a horizontal line) and where the curves tend. It also serves to estimate a value: start at the initial point and move along the segments, which is the idea of Euler's method.

The picture is reliable only when the existence and uniqueness theorem applies, because then each solution curve is unique and does not cross another. A frequent mistake is to draw segments of varying length: all should be short and equal, since only their direction matters.