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. $F(x,y,y',y'',y''',...)$

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. $F(x,y,y')$

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 $p_{n}(x)y^{(n)}+p_{n-1}(x)y^{(n-1)}+...+p_{0}(x)y=0$.

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 $f_{0}(x)+f_{1}(x)y+f_{2}(x)y'+f_{3}(x)y''+...f_{n}(x)y^{n}$. 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 $p_{n}(x)y^{(n)}+p_{n-1}(x)y^{(n-1)}+...+p_{0}(x)y=f(x)$, 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 $y'$, second order equations have $y'$ and $y''$, 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. $F(x,y,y',y'')$

separable equation

An equation where the x and y terms are multiplied and not added. ${\frac {dy}{dx}}=f(x)g(y)$

substitution method

A method of turning a non-separable equation into a separable one.