These are the calculations made using Mathematica 3.0 The results of these calculations are used in problem 3 on the second midterm.

```

Solve[{w-b^3*x-y==b^3*t,2*w+u*b^3*x-2*u*y==-1*b^3*t,a^3*x+y-a^3*z==0,
u*a^3*x-2*u*y-a^3*z==0},{w,x,y,z}]

-a^3 b^3 t + b^6 t - a^3 b^3 t u + b^6 t u + 2 a^3 b^3 t u^2 - 2 b^6 t u^2
w= - --------------------------------------------------------------------------
-2 a^3 + 2 b^3 + 4 a^3 u + 5 b^3 u-2 a^3 u^2 + 2 b^3 u^2

-3 (b^3 t + 2 b^3 t u)
x= - ----------------------------------------------------------
-2 a^3 + 2 b^3 + 4 a^3 u + 5 b^3 u - 2 a^3 u^2 + 2 b^3 u^2

3 a^3 b^3 t (-1 + u)
y= ---------------------------------------------------------
2 a^3 - 2 b^3 - 4 a^3 u - 5 b^3 u + 2 a^3 u^2 - 2 b^3 u^2

9 b^3 t u
z= --------------------------------------------------------------------------
2 a^3 - 2 b^3 - 4 a^3 u - 5 b^3 u + 2 a^3 u^2 - 2 b^3 u^2

Simplify[%]

b^3 (-a^3 + b^3) t (-1 - u + 2 u^2)
w= ----------------------------------------
-2 a^3 (-1 + u)^2 + b^3(2 + 5 u + 2 u^2)

-3 b^3 t (1 + 2 u)
x= ----------------------------------------
-2 a^3 (-1 + u)^2 + b^3(2 + 5 u + 2 u^2)

3 a^3 b^3 t (-1 + u)
y= ----------------------------------------
-2 a^3 (-1 + u)^2 + b^3(2 + 5 u + 2 u^2)

9 b^3 t u
z= ----------------------------------------
-2 a^3 (-1 + u)^2 + b^3(2 + 5 u + 2 u^2)

FullSimplify[%]

b^3 (-a^3 + b^3) t (-1 + u) (1 + 2 u)
w= -----------------------------------------
-2 a^3 (-1 + u)^2 + b^3 (2 + u) (1 + 2 u)

-3 b^3 t (1 + 2 u)
x= -----------------------------------------
-2 a^3 (-1 + u)^2 + b^3 (2 + u) (1 + 2 u)

3 a^3 b^3 t (-1 + u)
y= -----------------------------------------
-2 a^3 (-1 + u)^2 + b^3 (2 + u) (1 + 2 u)

9 b^3 t u
z= -----------------------------------------
-2 a^3 (-1 + u)^2 + b^3 (2 + u) (1 + 2 u)

```

For other mathematica calculation, the programs are called: ~thamlin/phy703/

```		2ndmid3.nb
2ndmid4.nb
2ndmid5.nb
```