Graphing differential equation solutions using a slope field

Once we have drawn a slope field, it is possible to graphically sketch the solution by hand, even if we do not have the formula for the solution. The solution to a differential equation is a family of functions, and this would be represented by a collection of curves passing through the slope field such that the curves are always parallel to the nearby slope lines.

For example, if we have the differential equation

$\frac{dy}{dx} = -xy$

We saw that the slope field would look like this

The completed slope field

If we draw a collection of curves for the solution to this differential equation, it would look like this

The completed slope field with a collection of solutions

Notice an important feature of the solution curves: they do not intersect with each other! There are a few situations in which we see exceptions to this rule, but in general this rule-of-thumb helps us to draw collections of curves in future problems.

If we are given a single point that a curve must pass through, in addition to a differential equation, we can also use the slope field to narrow down the collection of curves to a single one which would represent a solution.

For example, if we are told that

$\frac{dy}{dx} = -xy, \quad y(1) = 1.5$

then our solution to the differential equation would look like this

The completed slope field with a single solution passing through the point (1, 1.5)

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