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

538 Chapter 19 Ordinary differential equations I
Example19.4 Verifythaty =Ce x is a solution of dy =y, whereC is any constant.
dx
Solution If y = Ce x , then dy
dx = Cex . Therefore, for all values of x, dy = y and the equation
dx
is satisfied for any constantC;C is called an arbitrary constant and by varying it, all
possible solutions can be obtained. For example, by choosingC to be 1, we obtain the
solution of the previous example. Infact,y =Ce x isthe general solution of dy
dx =y.
More generally, to determine C we require more information given in the form of a
condition.Forexample,ifwearetoldthat,atx = 0,y = 4thenfromy =Ce x wehave
4=Ce 0 =C
so thatC = 4. Therefore y = 4e x is the solution of the differential equation which
additionally satisfies the condition y(0) = 4. This is called a particular solution. In
general, application of conditions to the general solution yields the particular solution.
Toobtainaparticularsolution,thenumberofgivenindependentconditionsmustbethe
same asthe number ofconstants.
Consider the following example.
Example19.5 Consider the second-order differential equation
d 2 y
dx 2 +y=0
The general solution of this equation can be shown tobe
y=Acosx+Bsinx
whereAandBarearbitraryconstants.
Find the particular solution which satisfies the conditions
(a) whenx = 0,theny = 0,and
(b) whenx = 3π 2 ,theny=1.
Solution Wenotethatbecausethegeneralsolutionhastwoarbitraryconstants,AandB,thentwo
conditions are necessary to obtain a particular solution. Applying the first condition to
the general solution gives
0=Acos0+Bsin0
=A
ThereforeA = 0,andthesolutionreducestoy =Bsinx.Applyingthesecondcondition
we find
1=Bsin 3π 2