Ordinary Differential Equations/Glossary

C

complementary function
The solution to the related homogenous equation for a nonhomogenous equation

D

differential equation
An equation with one or more derivatives in it. ${\displaystyle F(x,y,y',y'',y''',...)}$
domain
a solution of differential equation is a function y=(x)y which, when substituted along with its derivative among the differential equation satisfies the equation from all x in some specified interval.

F

first order equation
Any equation with a first derivative in it, but no higher derivatives. ${\displaystyle F(x,y,y')}$

G

general solution
The solution to a differential equation in its most general form, constants included

H

homogenous equation
Any equation that is equal to 0. In differential equations, its an equation ${\displaystyle p_{n}(x)y^{(n)}+p_{n-1}(x)y^{(n-1)}+...+p_{0}(x)y=0}$ .

I

initial condition
A value of a function or its derivative at a particular point, used to determine the value of constants for a particular solution
initial value problem
A combination of a differential equation and an initial condition. An initial value problem is solved for a total solution including the value of all constants
integration factor
A factor a differential equation is multiplied by to discover a solution .

L

linear equation
An equation who's terms are a linear combination of a variable and its derivatives. Such an equation is in the form ${\displaystyle f_{0}(x)+f_{1}(x)y+f_{2}(x)y'+f_{3}(x)y''+...f_{n}(x)y^{n}}$ . No terms for y or its derivatives may be raised to a power or placed inside a function such as sin or ln

N

nonhomogenous equation
Any equation that is not equal to 0. In differential equations, its an equation ${\displaystyle p_{n}(x)y^{(n)}+p_{n-1}(x)y^{(n-1)}+...+p_{0}(x)y=f(x)}$ , where f(x) is not 0.
non-linear equation
Any equation that is not a linear combination of a variable and its derivatives. Either one of the terms has the variable taken to a power, or is in a function such as sin or ln

O

O.D.E
See ordinary differential equation.
order
The highest derivative found in a differential equations. First order equations only have ${\displaystyle y'}$ , second order equations have ${\displaystyle y'}$  and ${\displaystyle y''}$ , etc.
ordinary differential equation
Any differential equation that has normal derivatives only

P

partial differential equation
Any differential equation that has partial derivatives in it
particular solution
A solution to a differential equation with all constants evaluated
P.D.E
See partial differential equation.

S

satisfy
to solve a differential equation. Used as an adjective, a solution to a differential equation satisfies that equation
second order equation
Any equation with a second derivative in it, but no higher derivatives. ${\displaystyle F(x,y,y',y'')}$
separable equation
An equation where the x and y terms are multiplied and not added. ${\displaystyle {\frac {dy}{dx}}=f(x)g(y)}$
substitution method
A method of turning a non-separable equation into a separable one.