Differential Equations Terms

Differential Equation

an equation containing the derivatives of one or more dependent variables, with respect to one or more independent variables

Ordinary Differential Equation

an equation that contains only ordinary derivatives of one or more dependent variables with respect to a single independent variable

Partial Differential Equation

an equation that involves partial derivatives of one or more dependent variables and of two or more independent variables

Order of a DE

the order of the highest derivative in the equation

Linear ODE

1) the dependent variable y and all its derivatives y', y'', y''',...,y^(n) are of the first degree, that is, the power of each term involving y is 1.

2) the coefficients a_0,a_1,...,a_n depend at most on the independent variable x

Nonlinear ODE

nonlinear functions of the dependent variable or its derivatives, such as sin(y) or e^y, cannot appear in a linear equation

Solution of an ODE

any function Φ, defined on an interval I and possessing at least n derivatives that are continuous on I, which when substituted into the nth-order ODE reduces the equation to an identity

Solution Curve

the graph of a solution of an ODE

Explicit Solution

a solution in which the dependent variable is expressed solely in terms of the independent variable and constants

Implicit Solution

G(x,y)=0 on interval I, provided that there exists at least one function Φ that satisfies the relation as well as the differential equation on I

One-Parameter Family of Solutions

a solution containing an arbitrary constant, c

N-Parameter Family of Solutions

an nth-order DE possesses an infinite nimber of solutions corresponding to the unlimited number of choices for the parameters

Particular Solution

solution of a DE that is free of arbitrary parameters

Singular Solution

solution that is not a member of a family of solutions of the equation, that is a solution that cannot be obtained by specializing any of the parameters in the family of solutions

Piecewise-defined Solutions

a particular solution, but cannot be obtained from the family of solutions by a single choice of c

System of ODEs

two or more equations involving the derivatives of two or more unknown functions of a single independent variable

Solution of a System of ODEs

a pair of differentiable functions, defined on a common interval I, that satisfy each equation of the system on this interval