7.4 Reasoning Using Slope Fields

Learning objectives

—FUN-7.C—Estimate solutions to differential equations—Topic 7.4• FUN-7.C Estimate solutions to differential equations.• FUN-7.C.3 Solutions to differential equations are functions or families of functions.• Enduring understanding FUN-7: Solving differential equations allows us to determine functions and develop models.

A slope field is not one solution curve. It displays the allowable tangent direction at many points, so many smooth curves can follow the field. A particular solution must pass through its given initial point and remain tangent to nearby segments as it moves across the plane.

Field feature near a solution	What it implies locally
positive segment slopes	the solution is increasing
negative segment slopes	the solution is decreasing
horizontal segments	the solution has derivative $0$ there
larger $\|\text{slope}\|$	the solution changes more steeply
a change from positive to negative slopes along the path	the curve turns from increasing to decreasing

For $dy/dx=y$ , segments above the $x$ -axis have positive slope, those on the axis are horizontal, and those below have negative slope. A solution through $(0,1)$ therefore rises as it follows the upper field; a solution through $(0,-1)$ falls in the lower field; $y=0$ itself follows the horizontal segments. Together these curves illustrate a family of solutions.

To estimate a solution through $(x_0,y_0)$ , begin exactly there and sketch smoothly in both directions, continually matching nearby segment slopes. The initial point narrows the field's whole family to the solution curve or curves compatible with that condition.

Do not connect every segment or assume a visible segment is part of the same solution. The field supports qualitative estimates of direction and shape; without additional analysis it does not supply an exact formula, exact distant values, or a guarantee that an initial condition has exactly one solution.

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—7.5 Approximating Solutions Using Euler’s MethodRead