Theory

Introduction

Each first-order differential equation leads to a family of curves, its integral curves (solution curves).

Conversely, each family of curves given by a single equation with a single parameter leads to a differential equation, and it is the equation whose solution curves are given by this family.

This correspondence

$$\text{family of curves}⟷\text{differential equation}$$

works in general, assuming the family of curves is provided by an equation (without derivatives) with a single varying parameter.

Introduction (part 2)

Something interesting we can do is define the orthogonal family of curves to a given family of curves. The orthogonal family is the family such that each curve is perpendicular to every curve of the original family.

For example, if one family of curves represents electric field lines, its orthogonal family represents equipotential lines. Or, if one family represents the level curves of a function, its orthogonal family represents the lines of steepest ascent.

The orthogonal family can be found from the original family by first computing the differential equation $\frac{dy}{dx}=f(x,y)$. The orthogonal family will be given by the differential equation $\frac{dy}{dx}=-1/f(x,y)$. (The orthogonal slope is the negative reciprocal slope.)