How does a field of small line segments reveal the family of solutions to a differential equation without solving it?
Interpret and sketch slope (direction) fields for first-order differential equations and sketch solution curves
WACE Specialist Unit 4 slope fields: reading the gradient at each point from a first-order differential equation, drawing short line segments, sketching solution curves that follow the field, and recognising equilibrium solutions, with a worked example.
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What this dot point is asking
SCSA wants you to read a slope field, construct one from a differential equation, and sketch solution curves through given points by following the field.
What a slope field shows
A first-order differential equation assigns a gradient to every point of the plane. The slope field is a grid of short line segments, each drawn with the gradient that the equation prescribes at that point. The field is a picture of the entire family of solutions at once, before any are found explicitly.
Reading and drawing the field
To draw the field, evaluate at sample points and draw a short segment of that gradient. Patterns help: if depends only on , all segments in a vertical line are parallel; if depends only on , all segments in a horizontal line are parallel.
Sketching solution curves
Where the segments are horizontal; a horizontal line on which this holds for all is an equilibrium (constant) solution that other curves approach or leave.