7.3 Sketching Slope Fields

Learning objectives

—FUN-7.C—Estimate solutions to differential equations• FUN-7.C Estimate solutions to differential equations.• FUN-7.C.1 A slope field is a graphical representation of a differential equation on a finite set of points in the plane.• FUN-7.C.2 Slope fields provide information about the behavior of solutions to first-order differential equations.• Enduring understanding FUN-7: Solving differential equations allows us to determine functions and develop models.

For a first-order differential equation $dy/dx=f(x,y)$ , a slope field places a short line segment at each selected point $(x,y)$ with slope $f(x,y)$ . Each segment gives the direction of any solution curve passing through that point; it is local information, not a separate piece of the solution.

Choose a finite grid of points.
Evaluate $f(x,y)$ at each point.
Draw a short centered segment with that slope: horizontal for $0$ , rising for positive, falling for negative.
To estimate a solution through an initial point, sketch a smooth curve that remains tangent to nearby segments.

For $dy/dx=x-y$ :

Point $(x,y)$	Slope $x-y$	Segment direction
$(0,0)$	$0$	horizontal
$(1,0)$	$1$	rising
$(0,1)$	$-1$	falling
$(1,1)$	$0$	horizontal

The zero-slope locations satisfy $x-y=0$ , or $y=x$ . Above that line, $x-y<0$ , so solution curves locally fall; below it, $x-y>0$ , so they locally rise. This pattern estimates how solutions bend as they move across the plane.

Do not connect the small segments end to end or treat all segments as one solution. A particular solution is a smooth curve through its specified initial point that is tangent to the field. A slope field estimates behavior; it does not by itself provide exact function values.

Next Topic

—7.4 Reasoning Using Slope FieldsRead