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MATH
[1251]

Definitions from Existence, uniqueness and linear operators

All definitions from Existence, uniqueness and linear operators.

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Lipschitz condition in the dependent variable

A function f(x,y)f(x,y) satisfies a Lipschitz condition in yy on a region when there is a constant MM such that, for each fixed xx and all y1,y2y_1,y_2 with (x,y1)(x,y_1) and (x,y2)(x,y_2) in the region,

∣f(x,y1)−f(x,y2)∣≤M∣y1−y2∣|f(x,y_1)-f(x,y_2)| \leq M|y_1-y_2|

The condition controls how quickly the slope field can change as yy changes. It is stronger than continuity in yy and prevents two solution curves through the same initial point from separating while they remain in the region.

  1. [1]:Local existence and uniqueness

Linear differential operator

On an interval II where its leading coefficient is nonzero, a linear differential operator of order nn can be divided by that coefficient and written in the normalised form

T[y]=y(n)+an−1(x)y(n−1)+⋯+a1(x)y′+a0(x)y.T[y] =y^{(n)}+a_{n-1}(x)y^{(n-1)} +\cdots+a_1(x)y'+a_0(x)y.

It satisfies

T[y1+y2]=T[y1]+T[y2],T[λy1]=λT[y1].T[y_1+y_2]=T[y_1]+T[y_2], \qquad T[\lambda y_1]=\lambda T[y_1].

Linearity follows by distributing derivatives and collecting terms. The constant-coefficient operator T[y]=y′′+ay′+byT[y]=y''+ay'+by is the operator used in second-order constant-coefficient equations and its characteristic equation.

  1. [1]:All forced solutions are an affine translate

Kernel of an operator

The kernel of a linear differential operator TT is

ker⁡(T)={y:T[y]=0}.\ker(T)=\{y:T[y]=0\}.

The kernel is exactly the set of homogeneous solutions. Adding an element of the kernel to a particular solution leaves the forcing unchanged. This is the differential-equation version of the kernel of a linear map: the kernel contains the inputs the operator sends to zero.

  1. [1]:Kernel
  2. [2]:All forced solutions are an affine translate

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Lipschitz condition in the dependent variableLinear differential operatorKernel of an operator