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)}.