Section 1.1
Basic Definitions and
Terminology
DIFFERENTIAL EQUATIONS
Definition: Adifferentialequation (DE)is
anequationcontainingthederivativesor
differentialsofoneormoredependent
variables,withrespecttooneormore
independentvariables.
PARTIAL DERIVATIVES
If
,
isafunctionoftwovariables,its
partialderivatives arethefunctions and
definedby
,
lim
,
→
lim
,
,
→
1
RULE FOR FINDING PARTIAL
DERIVATIVES OF z = f (x, y)
1. Tofind ,regard asaconstantand
, withrespectto .
differentiate
2. Tofind ,regard asaconstantand
differentiate
,
withrespectto .
CLASSIFICATION OF
DIFFERENTIAL EQUATIONS
Differentialequationsareclassifiedaccordingto
(i) type
(ii) order
(iii) linearity
CLASSIFICATION BY TYPE
Differentialequationsaredividedintotwotypes.
1. Anequationinvolvingonlyordinary
derivativesofoneormoredependent
variablesofasingle independentvariableis
calledanordinarydifferentialequation
(ODE).
2. Anequationinvolvingthepartialderivatives
ofoneormoredependentvariablesoftwo
ormore independentvariablesiscalleda
partialdifferentialequation (PDE).
2
CLASSIFICATION BY ORDER
Theorder ofadifferentialequationisthe
orderofthehighest‐orderderivativeinthe
equation.
CLASSIFICATION BY LINEARITY
Differentialequationsareclassifiedbylinearity
asfollows.
1. Ifthedependentvariable( )andits
derivativesareofthefirstdegree,and
eachcoefficientdependsonlyonthe
independentvariable( ),thenthe
differentialequationislinear.
2. Otherwise,thedifferentialequationis
nonlinear.
LINEAR DIFFERENTIAL
EQUATION
Adifferentialequationissaidtobelinear ifitcanbewritten
intheform
⋯
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SOLUTION OF A DIFFERENTIAL
EQUATION
Definition: Anyfunctionf definedonsome
intervalI,whichwhensubstitutedintoa
differentialequationreducestheequationto
anidentity,issaidtobeasolution ofthe
equationontheintervalI.
NOTE:Dependingonthecontextofthe
problemtheintervalI couldbeanopen
interval,aclosedinterval,ahalf‐openinterval,
oraninfiniteinterval.
AN n-PARAMETER FAMILY
OF SOLUTIONS
Whensolvingan th‐orderdifferentialequation
, , ,...,
0,weexpectasolution
, , ,...,
0 with arbitrary
parameters(constants).Suchasolutionis
calledan ‐parameterfamilyofsolutions.
PARTICULAR SOLUTIONS
Asolutionofadifferentialequationthatis
freeofarbitraryparametersiscalleda
particularsolution.Onewayofobtaininga
particularsolutionistochoosespecificvalues
oftheparameter(s)inafamilyofsolutions.
Aparticularsolutionthatcannot beobtained
byspecializingtheparametersinafamilyof
solutionsiscalledasingularsolution.
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