Terminology

A _____ _____ is an equation that contains the derivatives of one or more unknown functions/dependent variables with respect to one or more independent variables

Differential Equation

An _____ _____ _____ is a type of differential equation with only 1 independent variable

Ordinary Differential Equation

A _____ _____ _____ is a type of differential equation involving partial derivatives of two or more independent variables and functions

Partial Differential Equation

Functions are also called ____ ______

Dependent variables

_____ ______ is a way of writing derivatives like dy/dx

Leibniz Notation

_____ ______ is a way of writing derivatives like y’ or y’’

Prime Notation

____ _____ is the last way of writing derivatives by adding dots above the dependent variable

Dot Notation

_____ ______ is a way of writing partial derivatives, by adding subscripts representing the variable being differentiated

Subscript Notation

The _____ _____ _____ ______ ______ is the order of the highest derivative in the equation

Order of a differential equation

A ____ _____ ______ ______ is a differential equation with the order of 1, meaning the highest derivative present is the first derivative

First order differential equation

A ____ _____ ______ ______ is a differential equation with the order of 2, meaning the highest derivative present is the second derivative

Second order differential equation

The _____ _____ is a way of writing a first order DE, represented as M(x,y) dx + N(x,y) dy = 0

Differential Form

Assuming y is a dependent variable, the differential dy is defined in the equation:

dy = y’ dx

The ____ ______ of an ODE is F(x, y, y’, … y^n) = 0

General form

The ____ ______ of an ODE is d^n/ dx^n = f(x, y, y’,… y^n-1)

Normal Form

A ______ _____ ______ ______ is a differential equation that is linear in all dependent functions (y) and their derivatives (y’, y’’, etc.)

Linear ordinary differential equation

The image shows the _____ _____ for an ODE

Linearity Conditions

A _______ ______ _______ ______ is an ordinary differential equation that does not satisfy the linearity conditions

Nonlinear ordinary differential equation

A _____ ___ ____ ______ is a function that is continuous on an interval on all derivatives in the order of a DE, that, when substituted, reduces it to an identity

Solution to an ODE

The _____ of the solution is the interval of the independent variable in which the solution of the ODE works

Domain

The domain of the solution is also known as:

1) _____ ___ _____

2) _____ ___ ______

3) _____ ___ _______

Interval of Definition, Interval of Existence, Interval of Validity

A ____ _____ is a solution to an ODE that is identically 0 (zero) on an interval

Trivial Solution

A ______ ______ refers to the graph of a solution to an ODE

Solution Curve

An _____ ______ is a solution to an ODE in which the dependent variable is expressed only in terms of the independent variable and constants

Explicit Solution

An _____ _____ is a solution to an ODE that does not isolate the dependent variable and instead includes all variables in a mix

Implicit Solution

A ______ is a variable representing a constant in a DE, usually obtained with integration

Parameter

A _____ ______ _____ ____ ______ is a family of solutions to a DE parametrized by a single parameter and with an order of 1

One-Parameter family of solutions

An ____ _____ _____ _____ ______ is a family of solutions to a DE with order of n with n parameters

N-parameter family of solutions

A _____ ______ is a parameter given a specific numerical value

Particular Parameter

A _____ _____ is a solution to a DE with no parameters or with particular parameters (the parameters substituted by a number )

Particular Solution

A ____ _____ is a lone solution to an ODE outside of a family of solutions, regardless of the parameters used

Singular Solution

A _____ ____ ______ is a set of two or more ordinary differential equations (involving the derivatives of 2 or more unknown functions of a single independent variables) in the format:

-Dy/dt = f(t, x, y)

-dy/dt = g(t, x, y)

System of ODEs

The _____ ____ ____ _____ ____ ______ is a group of differentiable functions that satisfy the system of ODEs on the same given interval

Solution to a system of ODEs

An ____ _____ is a condition imposed on a solution to a DE at a given value

Initial Condition

An ____ _____ ____ is a problem asking you to find a solution to a DE that satisfies the Initial condition(s)

Initial value problem

A _____ _____ ______ is an initial value problem for a DE of nth order

Nth order IVP

A ____ ____ _____ is an initial value problem for a first order DE

First order IVP

A ____ ____ _____ is an initial value problem for a second order DE

Second order IVP

The _____ ___ ___ _____ _____ occurs when a function of x and y and the first partial with respect to y of this function are both continuous in a region R

Existence of a unique solution

A _____ ____ is a mathematical description of a system of phenomenon

Mathematical Model

The _____ ____ ____ _____ is the solution of a mathematical model for a system that involves time t

State of the System

A ____ ____ ____ is a chemical reaction DE whose rate is proportional to the concentration of a single reactant

First-Order Reaction

A ____ ____ _____ is a chemical reaction whose rate is dependent on one reactant squared or two separate reactants

Second-Order Reaction

Tan(a) is equivalent to _____

Dy/dx

A ____ ____ is a first order DE in normal form, or in the format dy/dx = f(x,y)

Slope Function

A slope function is also known as a ____ _____

Rate function

A ____ _____ is a line representing the slope magnitude at a specific point of a solution curve

Linear element

A ____ _____ or _____ _____ is a graphical representation of slopes at specified points using lineal elements

Slope field, direction field

An _____ ____ ____ ______ is an ODE of first order in which the independent variable does not appear explicitly

Autonomous 1st order ODE

____ _____ are values of the dependent variable y that make an autonomous function [ dy/dx = f(y)] equal 0

Critical Points

A ____ _____ is a critical point of an AUTONOMOUS DE, equivalent to a constant and a solution to the ODE

Constant solution

An _____ _____ is a critical point of an AUTONOMOUS 1ST ORDER ODE, equivalent to a constant and a solution to the ODE

Equilibrium Solution

A ____ ____ _____ _____ is a graphical representation of the trajectory of a dependent variable in an autonomous function, making the dependent variable the axis

One dimensional phase portrait

A _____ ____ is the vertical line in the phase portrait representing the dependent variable

Phase line

A _____ _____ is a function that is either always increasing or always decreasing

Monotonic Function

If a region for y of an autonomous DE is _____ ______ then it has a critical point that bounds it from above only

Bounded above

A ____ _____ is a region of y in an autonomous DE that is bounded by above and below

Bounded region