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

19 Ordinarydifferential
equationsI
Contents 19.1 Introduction 534
19.2 Basicdefinitions 535
19.3 First-orderequations:simpleequationsandseparationofvariables 540
19.4 First-orderlinearequations:useofanintegratingfactor 547
19.5 Second-orderlinearequations 558
19.6 Constantcoefficientequations 560
19.7 Seriessolutionofdifferentialequations 584
19.8 Bessel’sequationandBesselfunctions 587
Reviewexercises19 601
19.1 INTRODUCTION
Thesolutionofproblemsconcerningthemotionofobjects,theflowofchargedparticles,
heat transport, etc.,often involves discussion of relations of the form
d 2 x
dt 2 +6dx dt +2x=3t or dq
dt +8q=sint
In the first equation,xmight represent distance. For this case dx is the rate of change
dt
of distance with respect to timet, that is speed, and d2 x
represents acceleration. In the
dt2 second equation,qmight be charge and dq the rate of flow of charge, that is current.
dt
These are examples of differential equations, so called because they are equations involving
the derivatives of various quantities. Such equations arise out of situations in
whichchangeisoccurring.Tosolvesuchadifferentialequationmeanstofindthefunctionx(t)orq(t)whenwearegiventhedifferentialequation.Solutionstotheseequations
maybeanalyticalinthatwecanwritedownananswerintermsofcommonelementary
functions such as e t , sint and so on. Alternatively, the problem may be so difficult that
only numerical methodsare available, which produce approximate solutions.