order n constant coefficient differential equations and matrix exponential


Let P be a degree n>0 monic complex polynomial in one indeterminate, let f be a continuous functionMathworldPlanetmath on the real line, let k be an integer varying from 0 to n-1, and let yk be a complex numberPlanetmathPlanetmath. The solution to the ODE

P(d/dt)y=f(t),y(k)(0)=yk (1)

is

y(t)=ykgk(t)+0tgn-1(t-x)f(x)𝑑x, (2)

where gk(t) is the coefficient of zk in the product of P(z) by the singular part of

etzP(z).

Moreover, if A is a complex square matrixMathworldPlanetmath annihilated by P, then

etA=gk(t)Ak. (3)

(1) into

Y-BY=f(t)v,Y(0)=Y0 (4)

by putting Yk:=y(k), Y0k:=yk, and by letting B be the transposeMathworldPlanetmath companion matrixMathworldPlanetmath of P, and v the last vector of the canonical basis of n. The solution to (4) is

Y(t)=etBY0+0tf(x)e(t-x)Bv𝑑x.

There is a unique n-tuple of functions hk such that etA is the sum of the hk(t)Ak whenever A is a complex square matrix annihilated by P. The first line of Bk being the (k+1)-th vector of the canonical basis of n (for 0k<n), we obtain

y(t)=ykhk(t)+0thn-1(t-x)f(x)𝑑x,

so that the proof of (2) and (3) boils down to verifying

hk(t)=gk(t).

a real value of t, let G[X] be the sum of the gk(t)Xk, form the entire functionMathworldPlanetmath

φ(z)=etz-G(z)P(z),

multiply the above equality by P(z), and replace z by A.

Title order n constant coefficient differential equations and matrix exponential
Canonical name OrderNConstantCoefficientDifferentialEquationsAndMatrixExponential
Date of creation 2013-03-22 19:01:00
Last modified on 2013-03-22 19:01:00
Owner gaillard (1824)
Last modified by gaillard (1824)
Numerical id 7
Author gaillard (1824)
Entry type Definition
Classification msc 34-01