Principles of Mathematics in Operations Research

196 14 Special Transformations

Then, the unique solution is y(t) = etAyo +p(t), where

etAyo

2e* + e"

2e< - e~

and

p(t) =

/ 0 V (e~u + ue"") + e-'(e~" - ue-")] du

/o [e'(e-" + we"") + e^e"" + we" du

Then, after integration we have

p(t) =

-2 v(t)

3et-t

3e*-2

One can sotoe i/ie akwe differential equation system using Laplace trans-

forms:

y'(t) = Ay(t) + f(t)^s

<£>

s -1

-1 s

»i(s)~

*7i(«)

01

10

l

^i(s)

(*)

• 
  

i i

i

Then, the resolvent matrix is

(sI-A)-^

1

(s-l)(s + l

If we multiply both sides of (*) 6y (**), we /wroe

s 1

1 s

(**)

T](S) =

T)(s)

1

(s-i)(s + i) L

I

s-l +

3s+ 1

s + 3

1

y(*) =

• s^2 (s-l)(s + l)

1

s^2 + l

2s

0

-2

3e(-£

3e' -2

In order to find e , we expand right hand side of (•*) as

n(s) =

["1

2

1

L2

ll

2

1

1 i i 2 J

+ s + 1 •\ 2-

If we invert it, we will have the following

•" = £

e* + e * el — e *