# Definition:Integral Curve

## Definition

Let $\map f {x, y}$ be a continuous real function within a rectangular region $R$ of the Cartesian plane.

Consider the first order ODE:

- $(1): \quad \dfrac {\d y} {\d x} = \map f {x, y}$

Let $P_0 = \tuple {x_0, y_0}$ be a point in $R$.

The number:

- $\paren {\dfrac {\d y} {\d x} }_{P_0} = \map f {x_0, y_0}$

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

Let $P_1 = \tuple {x_1, y_1}$ be a point in $R$ close to $P_0$.

Then:

- $\paren {\dfrac {\d y} {\d x} }_{P_1} = \map f {x_1, y_1}$

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

Let $P_2 = \tuple {x_2, y_2}$ be a point in $R$ close to $P_1$.

Then:

- $\paren {\dfrac {\d y} {\d x} }_{P_2} = \map f {x_2, y_2}$

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 $\tuple {x, y}$ on $C$, the slope of $C$ at $\tuple {x, y}$ is given by $\map f {x, y}$.

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

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

Thus the general 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.

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 2$: General Remarks on Solutions