Principles of Mathematics in Operations Research

192 14 Special Transformations

homogeneous, otherwise it is non-homogeneous. If we assume 0 6/, and

2/(0) = 2/o, y'(0) = y'o,...,y(-n~1)(0) = y{^'X) where y 0 ,y' 0 ,... ,y 0 n~1] are n

(*)

specified real numbers, this is called initial value problems where y$ 's are the

prescribed initial values.

Example 14.1.2 (The 1*' and 2 nd order linear initial value problems)

y'(t) = a(t)y(t) + f(t), y(0) = y 0 ;

and for n — 2, the constant coefficient problem is

y"(t) + aiy'(t) + a 0 y(t) = b(t); y(0) = y 0 , y'(0) = y' 0.

Remark 14.1.3 Let

y(t) = yi(t) y' 1 (t) = y 2 (t)

y'(t) = y 2 (t) y' 2 (t) = y 3 (t)

,(«-!) (t) = y„(t) y'n(t) = -an-iyn{t)

<=> A =

0 1 0 ••

0 0 1 ••

0 0 0 ••

Ct 0 —Oil -012 • •

2/o =

2/o

2/o

(n-2)

2/0

(n-1)

L2/0 J

0

0

1

-«n-l.

. v(f) =

, /(*) =

0 '

0

0

b(t).

atiy 2 {t) - a 0 yi(t) + b(t)

2/1 (*)

2/2(0

yn-i(t)

yn(t)

We have linear differential systems problem:

y'(t) = Ay(t) + f{t); j/(0) = y 0.

14.2 Laplace Transforms

Definition 14.2.1 The basic formula for the Laplace transformation y to rj

is

n(s

Jo

*»(*) dt.

We call the function, n, the Laplace transform of y if Eteo € R 9 r](s) exists,
Vs > XQ. We call y as the inverse-Laplace transform of n.

V(s) = C{y(t)}, y(t) = C~l {V(s)}.