By Seymour Ginsburg

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**Extra info for Introduction to Mathematical Machine Theory**

**Example text**

First use transformation rules and then definitions. 5 Equations • Single Equations Equations (and differential equations) are indicated in Mathematica with a double equals sign, since the simple equals sign is already taken by definitions. In[84]:= Qul[84]= In[85]:= Qul[85]= a x +b == 1 b + a x == 1 myEquation =a x +b == 1 b + a x == 1 The Mathematica function for solving one or more equations is called Solve. It needs to know the equation and the variable. In[86]:= Qul[86]= Solve [a x + b == ~_-la+b}} {{X ~ 1, xl 52 Part 1 In[87):= Out[87)= Solve [myEquation, x] { {x --) - -1a+ b }} Let us give the solution a name: In[88):= Out[88)= result = % {{x --) _ -1a+ b }} The result of So 1 ve is written as a list of transformation rules, which may be irritating at first.

60 In[121]:= Out[121]= In[122]:= Part 1 Limit [ (x _ 1) 2 X2 -1 , x .... -1, Direction .... 1] , x .... -1, Direction .... -1 ] 00 Limit [ (x _ 1) 2 x 2 -1 Out[122]= Many other Mathematica functions can be manipulated in an analog way using options. These are always written as transformation rules. In Part 2 we will see many further examples with graphic functions. • Derivatives We have already seen the function D for the calculation of derivatives. Because it is used often, its name is (like N) one of the few exceptions in naming where only a letter is used in place of a whole word.

In[138]:= Out[138]= DSolve[{x' {{ x [ t I [t] +x[t] == 0, x[O] == 1, X' [0] == O}, x[t], t] 1 --) Cos [t 1} } As with algebraic equations, we can get the resulting functions as follows: In[139]:= Out[139]= x[t] /. %[[1]] Cos [t 1 Differential equations are also seldom solvable in closed form. In[140]:= DSolve[x' I [t] +Sin[x[t]] == 0, x, t] Solve: :verif : Potential solution {x[t] ~ComplexInfinity} cannot be verified automatically. Verification may require use of limits. Solve: :ifun: Inverse functions are being used by Solve, so some solutions may not be found.