1550078481-Ordinary_Differential_Equations__Roberts_

N-th Order Linear Differential Equations 211

c. Find the values of a which produce two equal real roots of the

auxili ary equation and write the general solution of (*) in each

case.

d. Find the intervals in which a produces two unequal real roots and

write the general solution of (*) in these intervals.

e. Find the interval in which a produces complex conjugate roots and

write the general solution of (*) for a in this interval.

decx

When c is a complex constant, dx = cecx. Consequently, the complex

function y(x) = ecx will be a solution of a homogeneous linear differential

equation with complex coefficients if and only if c is a root of the associated

auxiliary equation, just as is the case when the coefficients are all real.

In exercises 17-18 write the auxiliary equation, find its roots, and

then write the complex general solution of the given homogeneous

linear differential equation with constant complex coefficients.

17. y(^3 ) - (3 + 4i)y(^2 ) - ( 4 - 12i)y(l) + 12y = 0


18. y(^4 ) - (3 + i)y(^3 ) + (4 + 3i)y(^2 ) = 0

19. Consider the complex initial value problem (t) y' - iy = O; y(O) = l.

a. Show that YI(x) = eix satisfies the IVP (t) for all x in (-00,00).

b. Show that Y2 ( x) = cos x + i sin x satisfies the IVP ( t) for all x in

(-oo, oo).

c. Assuming the existence and uniqueness theorem stated in sec-

tion 4.1 applies to the IVP ( t) what can you conclude about YI ( x) =

e i x and y 2 (x) = cosx + isinx for all x in (-00, 00). (Note: This

exercise provides a different proof of Euler's formula.)

4.4 N onhornogeneous Linear Equations with

Constant Coefficients

Now let us consider then-th order nonhomogeneous linear differential equa-
tion

with constant coefficients an, an-I,.. ., aI, ao where an =f- 0 and b(x) is not

the zero function. Since the constants an, an-I,... , aI, ao are all continuous