Understanding Engineering Mathematics

The AE is

λ^2 + 2 λ+ 2 = 0

which has the complex roots

λ=− 1 ±j

This gives the two solutions

e(−^1 +j)x,e(−^1 −j)x

and the general solution

y=e−x(Aejx+Be−jx)

Now, using Euler’s formula (361

➤

)

e±jx=cosx±jsinx

we get
y=e−x(A(cosx+jsinx)+B(cosx−jsinx))
=e−x((A+B)cosx+j(A−B)sinx)
=e−x(Ccosx+Dsinx)

on writingC=A+BandD=j(A−B).
In this ‘real form’ thejis hidden and for physical problems whereyrepresents a real
quantity, such as distance or current, never resurfaces. However, as noted previously the
complex exponential form is sometimes useful in applications such as alternating current
theory.
In the above list of solutions we essentially have just three distinct types of functions:
eαx,xeαx,andeαxsinβx. Each of these functions has a particular sort of behaviour which
typifies the class to which it belongs, and the type of physical system it represents.

1. eαx gives an increasing (α>0) or decaying (α<0) exponential
   function:

x

y

eax a < 0 eax a > 0

Figure 15.1The functioneαx.

2. xeαxgives a similar type of function, but with a kink in it, and it also
   passes through the origin. Examples of this function were sketched in
   Chapter 10.


3. eαxsinβxgives either a simple oscillating wave (α=0), or a sinusoidal
   wave with amplitude that decreases (α<0) or increases (α>0) as