Basic Concepts of Differential Equations

A set of vocabulary flashcards covering the basic concepts and terms related to differential equations, their applications, and methods for solving them.

Differential Equation

An equation that contains the derivative of an unknown expression.

First Derivative

The term dy/dx, representing the rate of change of a dependent variable with respect to the independent variable.

Second Derivative

The term d²y/dx², representing the rate of change of the first derivative.

Independent Variable

The variable with respect to which differentiation is performed, found in the denominator of derivatives.

Dependent Variable

The variable that depends on the independent variable, found in the numerator of derivatives.

Order of a Differential Equation

The highest derivative of the dependent variable present in the equation.

Linear Differential Equation

A differential equation in which the dependent variable and its derivatives appear only to the first power and are not multiplied together.

Nonlinear Function

A function in which the dependent variable appears in powers higher than one or in products with its derivatives.

Integration

The process of finding the original function from its rate of change.

Separation of Variables

A method used when the rate of change depends on its current value and possibly on time or position.

Exponential Behavior

Describes processes such as decay, damping, relaxation, and diffusion.

Integrating Factor

A function used to rewrite a differential equation in a form that is integrable.

First-Order Linear Differential Equation

An equation of the form dy/dx + P(x)y = Q(x) where P(x) and Q(x) are known functions.

Mechanical Analogy

A comparison of differential equations to physical systems that respond proportionally and lose energy smoothly.

Resistance or Decay Term

The term P(x)y in a differential equation that represents energy or quantity loss from a system.