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

606 Chapter 20 Ordinary differential equations II
20.3 HIGHERORDEREQUATIONS
Inthissectionweshallconsidersecond-andhigherorderequationsandshowhowthey
can be represented as a set of simultaneous first-order equations. The main reason for
doingthisisthatwhenacomputersolutionisrequireditisusefultoexpressanequation
in this form. Details of the analytical solution of such systems are not considered here
although one technique isdiscussed inSection 21.10.
It is possible to express a second-order differential equation as two first-order equations.Thus
ifwehave
d 2 ( )
y dy
dx = f 2 dx ,y,x (20.6)
we can introduce the new dependent variablesy 1
andy 2
such thaty 1
= y andy 2
= dy
dx .
Equation (20.6) thenbecomes
dy 1
dx =y 2
dy 2
dx = f(y 2 ,y 1
,x) (20.7)
These first-order simultaneous differential equations are often referred to as coupled
equations.
Example20.1 Expressthe equation
d 2 y
dx 2 −7dy dx +3y=0
as a set of first-order equations.
Solution Lettingy 1
= y, andy 2
= dy
dx , we find dy 2
dx = d2 y
dx2. Therefore the differential equation
becomes
dy 1
dx =y 2
dy 2
dx −7y 2 +3y 1 =0
We note thatthese equations can alsobewritten as
dy 1
dx =y 2
dy 2
dx = −3y 1 +7y 2
or, inmatrix form,
( ) ( ) ( ) y
′
1
0 1 y1
=
−3 7 y 2
y ′ 2
where ′ denotes d dx .