Ordinary differential equations Basic notions Definition

Slope field

Builds on Ordinary differential equation Solution of an ordinary differential equation

Definition

Let $\Omega$ be an open subset of $\mathbb{R}^2$ and let $f : \Omega \to \mathbb{R}$ be a function. The slope field of the first-order equation $y' = f(x, y)$ is the map that assigns to each point $(x, y) \in \Omega$ the line through $(x, y)$ with slope $f(x, y)$ . In a picture, one draws a short segment of this line at each point of a grid in $\Omega$ .

Remarks

Let $u$ be a differentiable function on an interval. It is a solution of $y' = f(x, y)$ if and only if its graph is tangent to the slope field at each of its points. For $y' = y$ , the segments are horizontal on the $x$ -axis and get steeper as $|y|$ grows. For a system $x' = F(x)$ in the plane, the same picture with arrows is a picture of the vector field $F$ .

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Solution of an ordinary differential equation definition
Slope field definition

Tag	`0RKQQQACY7`
URI	`m2://0RKQQQACY7`
Aliases	`direction-field`
Kind	definition
Writing	W1 stated
MSC	34A26
Defines	Slope field
Uses	Ordinary differential equation, Solution of an ordinary differential equation

Slope field

Definition

Remarks

Reading path

Further reading

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