Understanding Engineering Mathematics

Sinceeλx =0wehave

aλ^2 +bλ+c= 0

Soλsatisfies a quadratic equation with the same coefficients as the DE itself,ay′′+by′+
cy. This equation inλis called theauxiliaryorcharacteristicequation (AE). As with all
quadratics with real coefficients (66
➤
) there are three distinct types of solution:

• Real distinct roots


• Real equal roots


• Complex conjugate roots

Each leads to a different type of solution to the DE. In each case we get two distinct
types of solution, and the general solution is formed from these. Below we summarise the
forms of these solutions. In each case you should verify the stated solutions by substituting
in the equations. The problems which follow will confirm the results in particular cases.

Roots ofAEreal and distinctα 1 ,α 2

This gives two solutions

eα^1 x,eα^2 x

and the general solution is then

y=Aeα^1 x+Beα^2 x

Roots real and equal,α

In this case two distinct solutions can be found:

eαx,xeαx

and the general solution is then

y=(Ax+B)eαx

Roots complex,α±jβ

This gives two solutions

e(α+jβ)x,e(α−jβ)x

giving a general solution

y=Ae(α+jβ)x+Be(α−jβ)x

=eαx(Aejβx+Be−jβx)

The imaginaryjis not always welcome here, so we useEuler’s formula(361

➤
) to put
the solution into real form

ejβx=cosβx+jsinβx

e−jβx=cosβx−jsinβx

y=eαx((A+B)cosβx+(A−jB)sinβx)