Principles of Mathematics in Operations Research

290 Solutions

Problems of Chapter 14

14.1

y"(t) - y(t) = e2t & s^2 r](s) - 2s - r)(s) = <& r?(s)(s^2 - 1) = r + 2s.
s-2 s-2

V(s) =

1

+

2s

(s-2)(s^2 -l) s^2 -l'

1. If V(s) = ^ + ^r + ^r == (»-2)(^-l)- (,-vU-iy Solve fOT A'B'C-

A+B+C=0

W + C = 0

-A + 25 - 2C = 1 *H-*4

C

=T-

Thus, ??(s) = ^TVT6(8-2) ^ + -- 6(3+1) 6(s-l)-

2. If V(a) = ^ + ^ = I^I=r) = ^j. Solve for tf,F:

£ + F = 2

E-F = 0

=»E = 1, F = 1.

Thus, i;(a) = -

Then, we have

• +• -+-
  1 ~ «+!•

^

=

3(^2)

+

6^TI)

+

2(^1) ^<>

=

r

2t+

h~*

+

?*•

14.2

2/0 + 1) = t/(fc) + 2efe <= zt](z)-z = r)(z) + (^2) rz <=> rj(z)(z-l) = z + 2
z — e z — e
V{z) = 2-1 (2-l)(,z-e)' + 2
If»?(«) = r-TTi v = -A + -s- then A = ^- and B = -V
Therefore, 'V / (2-lllz-ei z —^1 z —e^1 —e - ^
(z —l)(z —e) z —1 z —e e-1-
!/(*) =
2-1

• 2
  1 A 2
  1-e u-1; e+ — 1 \ z — e
  14.3
  «.j,(fc) = l + - (l-efc).
  1 — e
  ^ = -0.2y, ^ = -0.3s - O.ly, x(0) = 50, y(0) = 100.
  §-o-al.g~a4-<u4^-Iu»,+Iu2 = o<*)