1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 219

4.5 Initial Value Problems

The initial value problem for a homogeneous linear differential equation

with constant coefficients is defined as follows.

DEFINITION Initial Value Problem for a Homogeneous

Linear Differential Equation

The initial value problem for n-th order homogeneous linear dif-

ferential equations with constant coefficients consists of solving the

differential equation

where an, an-1, ... , ai, and ao are constants and where an =f. 0 subject to

the initial condit ions

(2)

where k 1 , k2, ... , kn are constants.

By the summary theorem stated at the end of section 4 .1, this initial value
problem has a unique solution on the interval (-oo, oo ). One way to solve the
initial value problem consisting of equatio ns (1) and (2) is to find the general
solution of (1), using a root finding routine when necessary. The general
solution of (1) will include n arbitrary constants, c1, c2, ... , Cn. The value of
these constants must be determined so that equations (2) are satisfied. This

requires the solution of a system of n linear equations in then unknowns, c 1 ,

C2, ... 'Cn.

The initial value problem for a nonhomogeneous linear differential equation
with constant coefficients is defined in the following similar manner.

DEFINITION Initial Value Problem for a Nonhomogeneous

Linear Differential Equation

The initial value problem for n-th order nonhomogeneous linear

differential equations with constant coefficients consists of solving the

differential equation