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

19.2 Basic definitions 537
Note also that the conditions for linearity are conditions on the dependent variable.
The linearity of a differential equation is not determined or affected by the presence of
non-linear terms involving the independent variable.
The distinction between a linear and a non-linear differential equation is important
because the methods of solution depend upon whether an equation is linear or nonlinear.
Furthermore, it is usually the case that a linear differential equation is easier to
solve.
Example19.2 Decide whether or notthe following equations are linear:
(a) sinx dy
dx +y=x
dx
(b)
dt +x=t3
d 2 y
(c)
dx 2 +y2 =0
dy
(d)
dx +siny=0
Solution In (a), (c) and (d) the dependent variable isy, and the independent variable isx. In (b)
the dependent variable isxand the independent variable ist.
(a) This equation islinear.
(b) Thisequationislinear.Itdoesnotmatterthatthetermint,theindependentvariable,
israised tothe power 3.
(c) This equation isnon-linear, the non-linearity arising through the termy 2 .
(d) This equation isnon-linear, the non-linearity arising through the termsiny.
19.2.3 Thesolutionofadifferentialequation
The solution of a differential equation is a relationship between the dependent and independent
variables such that the differential equation is satisfied for all values of the
independentvariable over a specified domain.
Example19.3 Verifythaty = e x isasolution ofthe differential equation
dy
dx =y
Solution If y = e x then dy
dx = ex . For all values of x, we see that dy
dx =yandsoy=ex isa
solution.Note alsothatthisequation isfirst order and linear.
Therearefrequentlymanydifferentfunctionswhichsatisfyadifferentialequation;that
is,therearemanysolutions.Thegeneralsolutionembracesalloftheseandallpossible
solutionscan beobtained from it.