MST30040 Differential Equations via Computer Algebra Fall 2010 Worksheet 1

Transcription

1 MST3000 Differential Equations via Computer Algebra Fall 2010 Worksheet 1 1 Some elementary calculations To use Maple for calculating or problem solving, the basic method is conversational. You type a command at the prompt (>) (followed by a semi-colon in older versions of Maple), press the enter or return key and Maple gives the output immediately below ! (sqrt(2) + 1) If any of the numbers in the input is a floating-point number (i.e. has a decimal point), then the output will usually be a floating-point approximation Compare the two following calculations In the first, Maple recognizes an integer and calculates using exact integer arithmetic. In the second, Maple views 31.0 as a floating-point number and therefore gives the result as a floating point approximation. The evalf function can be used to convert an exact number into a decimal floating point approximation. evalf ( ) 62 sin ( π ) 11 evalf ( sin ( π )) The accuracy in this floating-point number is 10 digits (depending on the particular release of Maple being used). This is the default accuracy in Maple 6 and later releases. The result can be obtained to any desired accuracy (within reason) by adding a second argument to evalf specifiying the number of digits required. 1

2 evalf ( sin ( ) ) π, 50 The default accuracy of the current version of Maple can be determined by typing Digits after the prompt: Digits Alternatively, the default accuracy can be changed as follows: Digits := 30 evalf ( sin ( )) π Digits 2 Assigning names to Maple objects The most recent output of a Maple computation can be referred to by a %. The second most recent output can be referred to by a %% % % 7 %% However, this device is not very useful if we want to refer to the same output in several subsequent computations or if we want to refer back to an output obtained much eariier in the Maple session. It is often convenient and generally a good idea to assign a name to the output of a computation or to any expression which may be used during a computation. You assign a name to an expression or output by using a := symbol; i.e. the syntax used is name:=expression. bob := 5 8 We now have a Maple object (in this case, an integer) called bob which can be manipulated at will. For instance, if we want to cube this number: bob 3 The name assigned to a Maple object can be any string of alphanumeric characters and underscores, subject to the condition that the first character is a letter. For example, answer, function3, a, bb 2, Bertie Ahern are valid names, while 3rd answer, 5a, Bush, x&y are not. However, certain names cannot be used for assignment since they have a predefined meaning in Maple. For example, the names Pi, set, Digits, I are reserved in Maple (the last is the symbol for the square root of -1). Compare the assignment operator := with the equal sign = : joe := x joe tom = x

3 tom The expression tom = x is an equation. eqn := tom = x eqn solve (eqn, x) solve (eqn, tom) 3 Functions and Expressions restart When working with Maple, or any computer algebra system, it is important to be aware of the type of object you are working with. For example, you should distingusih between exact numbers and floating point approximations. Another important distinction is between expressions and functions. An expression is just a meaningless combination of symbols, usually a mixture of algebraic operations and alphanumeric characters. The following are expressions: 5 x + 3 tom + joe 50 tom + 27 b 8 y 5 x + 7 y 3 = 2 + b The last expression is an equation, since it has an equal sign. The solve command usually accepts an equation as its input: solve ( 5 x + 7 y 3 = 2 + b, x ) A function is not an expression, it is an operation, algorithm or programme which accepts an input or inputs and returns an output. To define a function in Maple, the command format is input -> corresponding output (the input can be any symbol or expression). In order to work with a function we need to assign a name to it (unless it has already been assigned a name by Maple). For example, the cubing function: cube := a a 3 cube (10) cube (tom) cube (x) Observe carefully that applying the function cube to the expression x outputs the expression xˆ3. In this way we can convert a function into an expression. This process can be reversed using the unapply command: expr := x 2 3

4 expr (5) square := unapply (expr, x) square (5) square (x) Differentiation and Integration restart There are two differentiation operations in Maple, diff and D. The command diff is used to differentiate an expression with respect to a variable occurring in the expression and it outputs an expression. joe := x + 3 b 2 diff (joe, x) diff (joe, b) Second and higher derivatives of an expression can be obtained as follows: diff (joe, x, x) diff (joe, b, b) diff (joe, x, x, x) diff (joe, x$3) The differentiation operator, D, takes as its input a function (not an expression), and outputs a function: D ( x x 2) Or, better still, we can name the function once and for all, and then refer to it afterwards by name: tom := x x 2 D (tom) Note that tom is a function, not an expression; the same is true of its derivative. We may want to give a name to the derivative of tom : lisa := D (tom) Explain the following: diff (tom, x) Recall that tom(x) is now an expression involving x and thus can be differentiated using the diff command. The output will again be an expression involving x (and not a function). diff (tom(x), x) Most of the standard mathematical functions are built into Maple with reserved names: sin, cos, tan, sinh, exp, ln, log[10], sqrt, arcsin etc. Note that sin is a function, but sin(x) is the value of sin at the input x. sin ( ) π 2

5 log 10 (100) exp(1) D (sin) D (sinh) recip := D (ln) recip (5) Explain the outputs in the following: D (sin(x)) diff (sin, x) diff (sin(x), x) Explain the two following outputs. diff (sinx, x) diff (sin x, x) monster := sin ( tan ( log ( 1 + x 3 + sqrt(1 + x) ))) diff (monster, x) To find the third derivative of monster by hand would be a very formidable task, as the following Maple calculation shows: thirdderiv := diff (monster, x$3) Suppose now that we want to take this last output and evaluate it at 1.87, say. It helps that we have assigned a name - thirdderiv - to the output. However, the command thirdderiv(1.87) gets us nowhere: thirdderiv (1.87) This is because thirdderiv is an algebraic expression and not a function. Now let s make a function out of thirdderiv : harry := unapply (thirdderiv, x) We now have a function, which can be evaluated at 1.87: harry (1.87) Maple can handle partial derivatives of all order: bob := x 3 sin ( x y ) + sqrt ( x 3 + y 8) diff (bob, x) diff (bob, y) diff (bob, x, y) diff (bob, y, y) Maple can perform both definite and indefinite integration. Let us find the area under the curve y=sin(x) from x=0 to x=1: area := int(sin(x), x = 0..1) Maple tries to give an exact output if it can. It has done this here, but the result may not be what we want. We can use evalf to get a floating point approximation: 5

6 evalf (area) An alternative would have been to express (at least one of) the limits of integration as floating point numbers: int(sin(x), x = ) As we have seen, Maple can perform indefinite integration. It outputs one (usually the simplest) antiderivative of the given expression. The others, of course, are obtained by adding a constant: integ := int(x 7 sin ( x 2), x) gensol := integ + C diff (gensol, x) Sometimes Maple will simply output the original indefinite integral if it can find no simpler form for the antiderivative: int1 := int(sqrt(log(x)), x) diff (int1, x) Observe that Maple knows the Fundamental Theorem of Calculus: areafunction := t int(sqrt(log(x)), x = 1..t) diff (areafunction(t), t) Alternatively: D (areafunction) 5 The plot command restart The main command for graphing a function or list of functions of a single variable is plot. Compare the following: plot (sin) plot ([sin, cos]) plot (sin(x),x = 2..2) plot (sin(x),x = 0..10, y = 0..1) plot ([sin(3 x),cos ( x),sin(3 x) + cos ( x)], x = 0..5) Of course, we are free to the name the input variable as we choose: plot (sin(var),var = 3..3) The help page for any command can be accessed by typing?commandname after the prompt. Try the command?plot. 6