1.
a.

>> format rat
>> null(A)

ans =

  -467/1475    -959/1235        0      
   317/1187    1219/4905     -64/93    
   452/829    -1047/1814    -480/1901  
 -1086/1657    -181/7819   -1655/2656  
    94/3845    -140/45823   -497/23759 
  -355/1117     293/7377     251/923   


b.

>> C=[A,b1]

C =

  Columns 1 through 6 

      9            9           -8           -7           -4           -1      
      5            9           -2           -5           -6            9      
     -9           -9            8            7           -9            0      
    -18          -18           16           14          -31           -1      

  Column 7 

    -33      
    -17      
      0      
    -23      

>> rref(C)

ans =

  Columns 1 through 6 

      1            0          -3/2         -1/2           0         -33/13    
      0            1          11/18        -5/18          0          32/13    
      0            0            0            0            1           1/13    
      0            0            0            0            0            0      

  Column 7 

      0      
      0      
      0      
      1      


>> D=[A,b2]

D =

  Columns 1 through 6 

      9            9           -8           -7           -4           -1      
      5            9           -2           -5           -6            9      
     -9           -9            8            7           -9            0      
    -18          -18           16           14          -31           -1      

  Column 7 

     -4      
      4      
    -36      
   -112      

>> rref(D)

ans =

  Columns 1 through 6 

      1            0          -3/2         -1/2           0         -33/13    
      0            1          11/18        -5/18          0          32/13    
      0            0            0            0            1           1/13    
      0            0            0            0            0            0      

  Column 7 

   -46/13    
    58/13    
    40/13    
      0      

2.


>> F=[v1 v2 v3]

F =

     10           -9           -6      
     -5            7           -1      
      3           -1            9      
      0            3            9      
      8            6           -1      
      6            9            8      
      0            5           -8      

>> null(F)

ans =

   Empty matrix: 3-by-0


3.

>> G=[F, u]

G =

     10           -9           -6            2      
     -5            7           -1           -7      
      3           -1            9            9      
      0            3            9            7      
      8            6           -1            5      
      6            9            8            6      
      0            5           -8           -3      

>> rref(G)

ans =

      1            0            0            0      
      0            1            0            0      
      0            0            1            0      
      0            0            0            1      
      0            0            0            0      
      0            0            0            0      
      0            0            0            0      

>> H=[F,w]

H =

     10           -9           -6           54      
     -5            7           -1          -28      
      3           -1            9            2      
      0            3            9          -15      
      8            6           -1           13      
      6            9            8           -8      
      0            5           -8           -2      

>> rref(H)

ans =

      1            0            0            3      
      0            1            0           -2      
      0            0            1           -1      
      0            0            0            0      
      0            0            0            0      
      0            0            0            0      
      0            0            0            0      

4.b.

>> [t,y]=ode45('couple',[0,20],[-2,0,2,0]);
>> plot(t, y(:,1), '-', t, y(:,3), '-.')
>> title('Positions of Coupled Oscillators')
>> legend('First Mass', 'Second Mass')