Glossary
Differential Equation
An equation that relates an unknown function with one or more of its derivatives.
Example:
The equation dy/dx = ky is a common differential equation used to model exponential growth or decay.
Slope (m)
The measure of the steepness of a line or curve at a given point, representing the rate of change of the dependent variable with respect to the independent variable.
Example:
In the context of a slope field, the slope at the point (1,2) for dy/dx = x+y would be 1+2=3.
Slope Field
A graphical representation of the general solutions to a differential equation, where short line segments are drawn at various points, each having the slope indicated by the differential equation at that point.
Example:
When sketching a slope field for dy/dx = x, you'd draw horizontal segments along the y-axis and increasingly steeper positive slopes as x increases.
Visualize a solution
The process of graphically representing the family of functions that satisfy a given differential equation by sketching a curve that follows the direction indicated by the slope segments in a slope field.
Example:
By following the direction of the line segments in a slope field, you can visualize a solution curve that passes through a specific initial condition, even without finding the explicit function.