082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

566 Chapter 19 Ordinary differential equations I
Usingthe technique illustratedinExample 19.21 we obtain
x(t) =Acosωt +Bsinωt
whereA,Bareconstants. Note this solution may alsobe expressed as a singlewave
x =Ccos(ωt +φ)
whereC and φ are constants, using the technique described in Section 3.7.1, Combining
waves.
We note that this solution is sinusoidal and oscillates with time. It gives the position
of the mass at a given point in time. The constantsC and φ depend on the
position and velocity of the mass when it is released. These are known as theinitial
conditionsofthedifferentialequation(seeExample19.6onpage539).Itisintuitive
thatanequationthatdescribesthepositionofthemassatagiventimemusttakethese
into account.
Mathematically describing or modelling systems like this one using differential
equationsisanextremelyimportantdiscipline.Mathematicalmodelsofmechanical,
electricalandothersubsystemscanbelinkedtogetherandasaresultwholesystems
can be accurately characterized.
Example19.22 Givenay ′′ +by ′ +cy = 0,writedowntheauxiliaryequation.Iftherootsoftheauxiliary
equation arecomplex and aredenoted by
k 1
=α+βj
k 2
=α−βj
show thatthe general solution is
y(x) =e αx (Acosβx+Bsinβx)
Solution Substitution ofy = e kx into the differential equation yields
(ak 2 +bk+c)e kx =0
and so
ak 2 +bk+c=0
This isthe auxiliary equation. Ifk 1
= α + βj,k 2
= α − βj thenthe general solution is
y =Ce (α+βj)x +De (α−βj)x
whereC andDarearbitraryconstants. Usingthe laws ofindices thisisrewritten as
y=Ce αx e βjx +De αx e −βjx =e αx (Ce βjx +De −βjx )
Then, usingEuler’s relations, weobtain
y =e αx (Ccosβx+Cjsinβx+Dcosβx−Djsinβx)
=e αx {(C +D)cosβx+ (Cj−Dj)sinβx}