Ordinary Differential Equations/Glossary

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

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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.

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

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

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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.

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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 .

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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

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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

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O.D.E 
See ordinary differential equation.
order 
The highest derivative found in a differntial 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

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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.

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satisfy 
to solve a differential equation. Used as an adjective, a solution to a differential equation satisifes 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.

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