Scalar Differential Equations: Definitions

Front 1

Definition: Ordinary Differential Equation (ODE)

Back 1

An ODE is an expression which describes a relationship between a function of one variable and its derivative. The solution to a differential equation is a function which satisfies that relationship.

Front 2

Definition: Autonomous Differential Equation (Time-Independent)

Back 2

If the expression F(y,t) in the ODE (dy/dt) = F(y,t) does not specifically involve t, then it is an autonomous or time-independent differential equation.

Front 3

Definition: Non-Autonomous Differential Equation (Time-Dependent)

Back 3

If a differential equation specifically involves t (i.e. it must be written as (dy/dt) = F(y,t), then it is non-autonomous or time-dependent.

Front 4

Definition: First Order Differential Equation

Back 4

A differential equation which only involves the first derivative of the unknown function.

Front 5

Definition: ith Order Differential Equation

Back 5

An equation which involves derivatives up to and including the ith.

Front 6

Definition: Partial Differential Equations

Back 6

Partial differential equations describe a relationship between a function of several variables and its partial derivatives.

Front 7

Definition: Initial Value Problem

Back 7

Front 8

MC: The solution to a differential equation is: a) a function; b) a number

Back 8

a) The solution to a differential equation is a function.

Front 9

What is a differential equation that has a constant function as its solution called?

Back 9

1) a steady-state, 2) a stationary solution, 3) a stationary point, 4) a rest point, or 5) an equilibrium

Front 10

Definition: General Solution of a Differential Equation

Back 10

A parameterized solution y(t,k) of a differential equation dy/dt = F(y,t) is called a general solution if every solution of the differential equation can be achieved by letting k take on different values.

Front 11

Back 11

Front 12

How do you solve an ordinary differential equation?

Back 12

Step 1: Guess a solution
Step 2: Differentiate the solution with respect to t
Step 3: Check if derivative satisfies original differential equation

Front 13

Definition: Homogeneous Differential Equation

Back 13

Front 14

Definition: Non-Homogeneous Differential Equation

Back 14

Front 15

Separable Differential Equation

Back 15

The differential equation of the form dy/dx = f(x,y) is called separable, if f(x,y) = h(x)g(y); that is, if dy/dx = h(x)g(x).

Front 16

Back 16

Front 17

Back 17