Geometrical Significance of the Solutions of an Ordinary Differential Equation of the First Order

Since the primitive of an ordinary differential equation of the first order is a relation between the two variables $x$ and $y$ and a parameter $c$, the differential equation is said to represent a one-parameter family of plane curves. Each curve of the family is said to be an integral-curve of the differential equation.

Let the equation be

$\frac{d y}{d x}=f(x, y);$

let $D$ be a domain in the ( $x, y$ )-plane throughout which $f(x, y)$ is single-valued and continuous, and let $\left(x_{0}, y_{0}\right)$ be a point lying in the interior of $D$. Then the equation associates with $\left(x_{0}, y_{0}\right)$ the corresponding value of $d y / d x$, say $p_{0}$, and thus defines a line-element ¹ $\left(x_{0}, y_{0}, p_{0}\right)$ issuing from the point $\left(x_{0}, y_{0}\right)$. Choose an adjacent point $\left(x_{1}, y_{1}\right)$ on this line-element and construct the line-element $\left(x_{1}, y_{1}, p_{1}\right)$. By continuing this process a broken line is obtained which may be regarded as an approximation to the integral curve which passes through $\left(x_{0}, y_{0}\right)$.

This method of approximating to the integral-curves of a differential equation is illustrated in a striking manner by the iron filings method of mapping out the lines of force due to a bar magnet. Iron filings are dusted over a thin card placed horizontally and immediately above the magnet. Each iron filing becomes magnetised and tends to set itself in the direction of the resultant force at its mid-point, and if the arrangement of the filings is aided by gently tapping the card, the filings will distribute themselves approximately along the lines of force. Thus each individual filing acts as a line-element through its mid-point.

Let the bar magnet consist of two unit poles of opposite polarity situated at $A$ and $B$ and let $P$ be any point on the card. Then if the co-ordinates of $A, B$ and $P$ are respectively $(-a, 0),(a, 0),(x, y)$, if $r$ and $s$ are respectively the lengths of $A P$ and $B P$, and if $X, Y$ are the components of the magnetic intensity at $P$,

$ Y=\frac{y}{r^{3}}-\frac{y}{s^{3}}, \quad X=\frac{x+a}{r^{3}}-\frac{x-a}{s^{3}} $

The direction of the resultant force at $P$ is

$\begin{aligned} \frac{d y}{d x} & =\frac{Y}{X} \\ & =\frac{y}{x+a{\frac{r^3+s^3}{r^3-s^3}}}, \end{aligned}$

and this is the differential equation of the lines of force. Its solution is

$\frac{x+a}{r}-\frac{x-a}{s}=\text{const.}$

By giving appropriate values to the constant the field of force may be mapped out. The integral-curves are the lines of force approximated to by the iron filings.

Since it has been assumed that $f(x, y)$ is continuous and one-valued at every point of $D$, through every point there will pass one and only one integral-curve. Outside $D$ there may be points at which $f(x, y)$ ceases to be continuous or single-valued; at such points, which are known as singular points, the behaviour of the integral-curves may be exceptional.

Similarly, if an equation of the second order can be written in the form

$y^{\prime \prime}=f\left(x, y, y^{\prime}\right),$

where $f\left(x, y, y^{\prime}\right)$ is continuous and single-valued for a certain range of values of its arguments, the value of $y^{\prime}$ at the point $\left(x_{0}, y_{0}\right)$ can be chosen arbitrarily within certain limits, and thus through the point $(x_{0}, y_{0})$ passes a one-fold infinity of integral-curves. The general solution involves two arbitrary constants, and therefore the aggregate of integral-curves forms a two-parameter family.

In general the integral-curves of an ordinary equation of order $n$ form an $n$-parameter family, and through each non-singular point there passes in general an $(n-1)$-fold infinity of integral-curves.

Footnotes

1. The line-element may be defined with sufficient accuracy as the line which joins the points ( $x_{0}, y_{0}$ ) and ( $x_{0}+\delta x, y_{0}+\delta y$ ) where $\delta x$ and $\delta y$ are small and $\delta y / \delta x=p_{0}$. ↩