Definition:Integral Curve

Definition
Let $f \left({x, y}\right)$ be a continuous real function within a rectangular region $R$ of the Cartesian plane.

Consider the first order ODE:
 * $(1): \quad \dfrac {\mathrm d y} {\mathrm d x} = f \left({x, y}\right)$


 * IntegralCurve.png

Let $P_0 = \left({x_0, y_0}\right)$ be a point in $R$.

The number:
 * $\left({\dfrac {\mathrm d y} {\mathrm d x} }\right)_{P_0} = f \left({x_0, y_0}\right)$

determines the slope of a straight line passing through $P_0$.

Let $P_1 = \left({x_1, y_1}\right)$ be a point in $R$ close to $P_0$.

Then:
 * $\left({\dfrac {\mathrm d y} {\mathrm d x} }\right)_{P_1} = f \left({x_1, y_1}\right)$

determines the slope of a straight line passing through $P_1$.

Let $P_2 = \left({x_2, y_2}\right)$ be a point in $R$ close to $P_1$.

Then:
 * $\left({\dfrac {\mathrm d y} {\mathrm d x} }\right)_{P_2} = f \left({x_2, y_2}\right)$

determines the slope of a straight line passing through $P_2$.

Continuing this process, we obtain a curve made of a sequence of straight line segments.

As successive points $P_0, P_1, P_2, \ldots$ are taken closer and closer to each other, the sequence of straight line segments $P_0 P_1 P_2 \ldots$ approaches a smooth curve $C$ passing through an initial point $P_0$.

By construction, for each point $\left({x, y}\right)$ on $C$, the slope of $C$ at $\left({x, y}\right)$ is given by $f \left({x, y}\right)$.

Hence this curve is a solution to $(1)$.

Starting from a different point, a different curve is obtained.

Thus the solution to $(1)$ takes the form of a set of curves.

This set of curves are referred to collectively as integral curves.

Also see

 * Picard's Existence Theorem which provides a rigorous analysis of the above informal argument.