1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 207

Using the computer program POLYRTS, we find the roots of this polynomial

are -3, -1, -1, -1, -1, 0, 0, 0, 2, and 2. That is, -3 is a root of multiplicity

one, -1 is a root of multiplicity four, 0 is a root of multiplicity three, and

2 is a root of multiplicity two. Corresponding to the root -3, we have the

solution

Corresponding to the root -1 of multiplicity four, we have the four linearly

independent solutions

y3 = xe -x ,

Corresponding to the root 0 of multiplicity three, we have the three linearly

independent solutions

And corresponding to the root 2 of multiplicity two, we have the two linearly

independent solutions

yg = e^2 x and Y10 = xe^2 x.

The set {y 1 , yz, ... , y 1 o} contains ten linearly independent solutions on the

interval (-00,00) of the DE (10). Therefore, the general solution of the
DE (10) is

Y = C1Y1 + c2y2 + c3y3 + · · · + c10Y10

= c1e-^3 x + Cze-x + C3Xe-x + C4X^2 e-x + C5X^3 e-x + C61 + C7X + csx^2 +

Cge2x + C10Xe2x

where c 1 , c 2 , ... , c10 are arbitrary constants.

Complex Roots Recall from calculus that the Taylor series expansions

about x = 0 for ex, cos x, and sin x are

oo (-l)nx2n x2 x4 x6

cosx = :L ( 2 n)! = 1- 21 + 41 - 61 + ...

n=O