1550078481-Ordinary_Differential_Equations__Roberts_

562 Ordinary Differential Equations

Exercises 9.2 Pendulum Systems

7. a. Let u1 = Y1, u2 = y~, u3 = y2, and U4 = Y2· Then

0 1 0 0

u' 1 U1

( g m2 ) 0 m2 0

u' 2 - £1 + m1£1 m1£1 U2

u~^0 0 0 1 U3

u' 4 g m2 ( g m2 ) U4

-+-- (^0) - £2 + m1£2 0
£2 m1£2

7. b. u = c1[(cos,8x)v1 - (sin,8x)v2] + c2[(sin,8x)v 1 + (cos,8x)v2]
   
   
   • C3 [ (cos /X )v3 - (sin /X )v 4] + c4[(sin /X )v3 + (cos /X )v 4]
   
   
   
   

where ,8 = 4.42568, / = 6.26525 and

V1 = (~m) ' V 2 =

.0000000

(

.000000)

• .176 837
  .000000 '


• .353 193

(

-.000152981)

0.000000

v^3 = 0.2253 42 ' and

0.000000

(

0.00000 )

• -0.00095846 3
  V^4 - 0.00000
  1.41182

Exercises 9.4 Mixture Problems

9. 
   

   a. q c -- cie - .0633975t (..590690 806898) + c (^2) e -.236603t (-.9 77416 .357759)

   
   


10. 
    

    b. q = q c + G~~)

    
    

9. c. q = -221.49499e-·^0633975 t (:~~)

+^31 ·^542932 e -.236603t (-.35.9 77416 7759) + (200) 100