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Diff Eq

definitions for differential equations terms

IVP A differential equation with initial values is called an initial value problem: y(x_0)=c_0,y'(x_0)=c_1,...,y^(n-1)(x_0)=c_(n-1)
Homogeneous Equation If f(&x,&y)=&^kf(x,y), then f is said to be homogeneous of degree k, where k is a real number.
differential equation An equation that contains derivatives of differentials of one or more variables is said to be a differential equation.
dependent variable A dependent variable is a function of an independent variable (allowed to vary on its own).
parameters Parameters are quantities allowed to vary beside the variables, but are not either independent or dependent.
order of diff eq The order of a differential equation is the order of the highest-ordered derivative in the equation (not the degree).
general form of nth order diff eq F(x,y,dy/dx,d2y/dx2,...,dny/dxn)=0
solution of diff eq y_1 is a solution of F(x,y,...,dny/dxn)=0 if: (a) y_1 is n-times differentiable (b) F(x,y_1,...,dny_1/dxn)=0
explicit solution A solution in explicit form is called an explicit solution (the dependent variable is written explicitly as a functioin of the independent variable(s) only).
implicit solution g(x,y)=0 is an implicit solution if it defines at least one real explicit solution to the differential equation on a given interval
particular solution A particular solution is a solution without any arbitrary constants; these arbitrary constants are plugged in for.
general solution A general solution is a solution form which every solution can be derived (by choosing arbitrary constants).
singular solution A solution that cannot be obtained by giving values to arbitrary constants in a family of solutions is called a singular solution
autonomous equation y'=f(y) where f, the driving function, is a function of the dependent variable only is called an autonomous equation. The independent variable does not appear explicitly.
Created by: efgray