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.