Principles of Mathematics in Operations Research

Solutions 291

dt^2

= s^2 r](s) - sy(0) - y (0) = s^2 rj(s) - 100s + 25,

since %\t=0 = -0.3(50) - 0.1(100) = -25. Moreover,

V

&y_

dt

sr)(s) - j/(0) = srj(s) - 100

(*) : [s^2 rj{s) - 100s + 25] - 0.06??(s) + 0.1[s?y(s) - 100] = 0

<£> v(s)(s^2 - 0.06 + 0.1s) = 100s - 25 + 10 &•

100s - 15

ri(s) =

A B

(s + 0.3)(s-0.2) s + 0.3 s-0.2 +

A + B = 100, -0.24 + 0.3S = -15 => 4 = 90, i5 = 10 =>

y(t) = 90e-°-3t + 10e°-2t

,., , , 90 10

s + 0.3 s-0.2

r 2/(o)i

2/(1)

2/(2)

2/(3)

L 2/(4) J

=

"100.0000"

78.88767

64.31129

54.81246

49.36289

14.4

Let x(n) — Fn+i, and the initial conditions are x(0) = 1, x(l) = 1.

x(n + 1) = x(n) + x(n - 1), n - 2, 3,... (¥)

r][x(n + 1)] = zrj(z) — zx(0) — zr)(z) - z and rj[x(n - 1)] = \r){z).

(¥) : x(n + 1) = x(n) + x(n - 1) o- zrj(z) - z - rj(z) r)(z) = 0

<*ri{z)

z-l-± z{l-±-\)

z ^ z zJ /

•«•

7]{z) =

z z^

1 A

l _ l +

£

(l - ^) (l - ^) ^^ 1

4 + 5 = 1

1+75

22

,M)+Bfii^]=0^=^5=^^

2*

(¥) : V(z)

2z

5 + Vb ,

10 I

10 10

5-\/5 / 1_

!_i^i; io ^i_^

Since Z (yrs-) = any(n), we have