We study a number of ways that families of curves can be defined using differential equations.

Applications to Curves

One-Parameter Families of Curves

We begin with two examples of families of curves generated by varying a parameter over a set of real numbers.

Equations (eq:4.5.1) and (eq:4.5.2) define one–parameter families of curves. (Although (eq:4.5.2) isn’t in the form (eq:4.5.3), it can be written in this form as .)

Recall from Trench 1.2 that the graph of a solution of a differential equation is called an integral curve of the equation. Solving a first order differential equation usually produces a one–parameter family of integral curves of the equation. Here we are interested in the converse problem: given a one–parameter family of curves, is there a first order differential equation for which every member of the family is an integral curve. This suggests the next definition.

To find a differential equation for a one–parameter family we differentiate its defining equation (eq:4.5.5) implicitly with respect to , to obtain

If this equation doesn’t, then it’s a differential equation for the family. If it does contain , it may be possible to obtain a differential equation for the family by eliminating between (eq:4.5.5) and (eq:4.5.7).

The next example shows that members of a given family of curves may be obtained by joining integral curves for more than one differential equation.

Geometric Problems

We now consider some geometric problems that can be solved by means of differential equations.

The following interactive will allow you to explore a family of curves with an interesting property. To explore, change the values of and . What do you observe about the relationship between the -coordinate of the point of tangency and the -intercept of the tangent line?

The following example shows how to find a family of solutions that posses the property you observed.

Orthogonal Trajectories

Two curves and are said to be orthogonal at a point of intersection if they have perpendicular tangents at . (See figure below).

A curve is said to be an orthogonal trajectory of a given family of curves if it’s orthogonal to every curve in the family. For example, every line through the origin is an orthogonal trajectory of the family of circles centered at the origin. Conversely, any such circle is an orthogonal trajectory of the family of lines through the origin.

Orthogonal trajectories occur in many physical applications. For example, if is the temperature at a point , the curves defined by

are called isothermal curves. The orthogonal trajectories of this family are called heat-flow lines, because at any given point the direction of maximum heat flow is perpendicular to the isothermal through the point. If represents the potential energy of an object moving under a force that depends upon , the curves (eq:4.5.16) are called equipotentials, and the orthogonal trajectories are called lines of force.

From analytic geometry we know that two nonvertical lines and with slopes and , respectively, are perpendicular if and only if ; therefore, the integral curves of the differential equation are orthogonal trajectories of the integral curves of the differential equation because at any point where curves from the two families intersect the slopes of the respective tangent lines are

This suggests a method for finding orthogonal trajectories of a family of integral curves of a first order equation.

Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)