Unit 7: Differential Equations - Formulation and Visualizations

Differential Equation (DE)

An equation containing one or more derivatives of an unknown function.

Rate of Change

Corresponds to a derivative; represents how fast a quantity changes with respect to time.

Constant of Proportionality (k)

In differential equations, indicates whether growth is increasing (k > 0) or decay (k < 0).

General Solution

Contains an arbitrary constant + C, represents a family of functions.

Particular Solution

A specific function derived from the general solution using an Initial Condition.

Derivative

A measure of how a function changes as its input changes.

Newton's Law of Cooling

The rate of change of a temperature is proportional to the difference between the object’s temperature and ambient temperature.

SLOPE FIELD

A graphical tool used to visualize the general solution to a first-order differential equation.

Initial Condition

A specified value that helps determine a particular solution from the general solution.

Logistic Growth

A model where the growth rate of a population is proportional to the product of its current population and the difference from its maximal capacity.

Equilibrium Solutions

Solutions to a differential equation where the derivative is zero, indicating no change.

Flow

The path traced by a solution curve in a slope field.

Slope Segment

A short line drawn at a point in a slope field representing the derivative's value at that point.

Unicity Theorem

States that in a slope field, solution curves cannot cross each other.

Verification Process

Steps taken to check if a proposed function satisfies a given differential equation.

Mathematical Model

An expression that represents a real-world situation using mathematical language and symbols.

Substitution

The act of replacing variables in a differential equation with their corresponding expressions.

Differentiable

A property of a function indicating that it has a derivative at all points in an interval.

Constant of Proportionality (k)

The value indicating how one variable responds to another in a proportional relationship.

Ambience Temperature (A)

A constant temperature that affects the rate of change in Newton's Law of Cooling.

Horizontal Asymptote

A line that a function approaches as the independent variable approaches infinity.

Initial Value Problem

A differential equation along with a specified value of the dependent variable at a specific value of the independent variable.

Flow Path

The trajectory followed by a solution curve that illustrates continuity in a slope field.

Undefined Slopes

Points in a slope field where derivative calculations result in division by zero, leading to no slope representation.

Integer Coordinates

Coordinate points consisting of whole numbers, often used in constructing slope fields.

Graphs Compatibility

Matching equations to their corresponding slope field graphs in problem-solving situations.

Testing Validity

The process of checking if a function works as a solution to a differential equation.