A Closer Look at Second Order Differential Equations. Duncan Lawson. BP Mathematics Centre. Coventry University. Coventry CV1 5FB. Great Britain. Abstract.
Computer Algebra and the Mathematics Curriculum: A Closer Look at Second Order Differential Equations Duncan Lawson BP Mathematics Centre Coventry University Coventry CV1 5FB Great Britain Abstract There is no longer any question about whether the availability of computer algebra packages will lead to changes in the mathematics curriculum. The question is now in what ways will the curriculum change. The question is of particular importance in higher education where many students, particularly those studying engineering, have traditionally been taught mathematical methods courses primarily from the point of view of techniques. The question is already being asked by many in the engineering community if, given the mathematical power now available at the push of a button, there is still a need for these kind of mathematics courses within engineering degrees.
In this paper rather than discussing the whole of the mathematics curriculum for engineers we will focus on just one topic: second order differential equations. We shall examine a number of ways in which students could study this topic both with and without the aid of computer algebra. We shall identify advantages and disadvantages of the various approaches. We conclude that topics like this can be used to provide insight into over-arching mathematical themes such as linearity which are of enormous importance throughout the whole of engineering. This macroscopic picture is something which has often been missing from the traditional technique oriented approach. 1. Introduction The impact of computer algebra systems on the mathematics curriculum has been envisaged for a number of years. In 1981 Stern [1] predicted that computer algebra systems ‘... will do for mathematical analysis what the hand calculator has done for calculation and arithmetic.’ This prophecy has not yet been fulfilled. Although computer algebra systems are enhancing the mathematical learning experience of many students, the usage of such software is nowhere near as widespread as that of the hand calculator. Broadly speaking, the impact of computer algebra on mathematical education has been evolutionary rather than revolutionary. A major factor in delaying the revolution has been the issue of access. Whilst hand calculators can be carried with students and so be available to them at all times until recently the same has not been true of computer algebra systems. Students have required a PC to access the software. Even though a small but growing number of 1
students own a computer they do not have this with them at all times. However, there is now a hand held ‘calculator’ (Texas Instruments TI-92) with computer algebra capabilites more than sufficient for most engineering undergraduates available for around £150. Using past experience with technology as a guide it seems reasonable to assume that this price will drop and all students will soon have the same instant access to computer algebra that they currently have to numerical calculation. The accessibility of computer algebra will force the educational establishment to react. Already in Britain the authorities are giving consideration to the introduction of a ‘technology-free’ examination paper in A level mathematics. (A levels are British national examinations taken by students usually at age 18 and are used by most universities as an entry requirement.) There will however be other papers which are not ‘technology-free’. In these papers the nature of the questions which can reasonably be set must change. There is little point in setting a question which, for example, requires a candidate to differentiate xsin2x if, rather than explicitly using the product rule, the candidate simply presses a few buttons on a sophisticated calculator. Ten years ago Calmet [2] foresaw that computer algebra would lead some (particularly those with responsibilities for budgets) to ask the question ‘Are mathematics teachers still needed in engineering curricula?’ Whilst almost all mathematicians would give the resounding answer ‘yes’ to this question there is a danger that others will reach the opposite conclusion. Mathematicians may remain conservative and retain traditional approaches with heavy emphasis on manipulation and technique with little attention given to engineering application. However, users of mathematics may then reasonably argue that their students can be proficient manipulators and appliers of techniques simply by learning to push the right buttons. In the past it was essential that students had highly developed manipulatory skills and were taught a wide range of techniques. There was no alternative. However, with the advent of computer algebra technology there is an alternative and it is in everyone’s interests that the benefits offered by the alternative are taken advantage of. In the remainder of this paper we will look closely at the topic of second order differential equations. This is a subject of some importance to engineers of various disciplines because of its use to represent oscillations of mechanical and electrical systems. We will discuss four alternative ways to explore this topic highlighting the advantages and disadvantages of the various approaches. We shall use the sample problem given below as an illustrative example of these different learning methods.
Sample Problem This problem is taken from Mustoe [3] p 395. Find the solution of the differential equation d2y dy + 2 + 4 y = cos(2 t ) 2 dt dt 2
given the initial conditions that when t = 0 then y = 0 and y’=0 . 2. The Algorithmic Approach - Mark 1 In this approach students are taught that there are a number of stages which must be routinely followed to solve this kind of differential equation. These can be listed as: 1. write down and solve the auxiliary equation; 2. determine the complementary function from the solutions of the auxiliary equation; 3. by using a suitable trial function with undetermined coefficients calculate a particular integral; 4. formulate the general solution from the sum of the complementary function and the particular integral; 5. use the initial conditions to determine the values to be given to the arbitrary constants in the general solution. For the sample problem these steps are shown below. m2 + 2m + 4 = 0 .
The auxiliary equation is
The solutions of this equation are The complementary function is thus
−1 ± j 3 .
(
ycf = e − t A cos( 3t ) + B sin( 3t )
)
y = C cos( 2 t ) + D sin( 2 t ) as a trial function to find a particular integral. is substituted into the differential equation we find that 1 4 D cos(2 t ) − 4C sin(2 t ) = cos(2 t ) . From which we deduce that C = 0 and D = . 4
We choose When this
This gives
y pi =
1 sin(2 t ) 4
The general solution is then obtained by summing the complementary function and the particular integral
(
)
1 y = e − t A cos( 3t ) + B sin( 3t ) + sin(2 t ) 4 To find a particular solution we use the initial conditions to determine values for the constants A and B. The condition on y gives A = 0 and (after differentiating to find −1 − 3 y’) the second condition gives B = = . 6 2 3 The solution of the problem is thus y=
1 3 −t sin(2 t ) − e sin( 3t ) 4 6
3
Standard textbooks (for example Mustoe [3], Attenborough [4]) and presumably most lectures which describe this algorithmic approach to solving second order linear constant coefficient equations at least show the origins of the auxiliary equation by substituting emt into the differential equation. The textbooks also usually include a brief discussion of the linearity of the differential operator. However, the tasks assigned to students (including for assessment) are usually the solution of a number of problems like the one outlined above. In such circumstances the ability to solve this kind of problem can be perceived by the student to be the most important feature of this topic. This can lead to the centrality of key concepts such as linearity being overlooked. This algorithmic approach is consistent with the aim of producing skilled manipulators who are able to solve a given class of problem. By asking students to solve a number of problems they can be exposed to the full range of solution types. They are also given ample opportunity to practice their manipulatory skills. Following the algorithm as a recipe which, if applied without algebraic error, will produce the correct answer does not emphasise the importance of key mathematical principles neither does it give insight into the engineering phenomena that the differential equation purports to represent. Given the volume of algebra required students are prone to focus on the detail of each step and to lose sight of the overall picture. 3. The Algorithmic Approach - Mark 2. This approach is very similar to the one outlined in the previous section, with the exception that students use a computer algebra package to carry out the manipulation involved at each step. A MAPLE worksheet which shows the required instructions with the output produced is shown in Figure 1 below. The disadvantages of this approach are similar to that of the algorithmic approach mark 1. It is however different in that it does not give students the opportunity to practice their algebraic skills. On the other hand students are less likely to be hindered by algebraic errors. Also, the time required to solve a problem is greatly reduced. This means that students can investigate a larger number of problems than using the previous approach and (hopefully) have time for reflection.
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Figure 1: MAPLE worksheet to implement algorithmic approach 4. The Minimalist Approach In this approach the computer algebra system is used to find the solution to the problem with as little intervention by the user as possible. In MAPLE it is possible to find a solution with a single command as shown in Figure 2. dsolve({diff(y(t),t,t)+2*diff(y(t),t)+4*y(t)=cos(2*t),y(0)=0,D(y)(0)=0},y(t)); Figure 2: MAPLE worksheet showing single command solution. If all that students are taught about second order differential equations is that they can be solved by a command like this then it is debatable just what they learn beyond computer algebra syntax. It may appear that this approach has something to commend it if all that is required of engineering undergraduates is that they can obtain the solution and then interpret it in an engineering context. However, there is a hidden difficulty with this kind of approach. If MAPLE has not been instructed to use trigonometric identities to 5
simplify expressions the solution that MAPLE gives to the differential equation appears to be considerably more complicated than the simple form above, as shown in Figure 3 below. dsolve({diff(y(t),t,t)+2*diff(y(t),t)+4*y(t)=cos(2*t),y(0)=0,D(y)(0)=0},y(t));
combine(“,trig);
Figure 3: Unexpected Solution from MAPLE In order to have a solution in a usable form the student must be aware that the solution need not look like this. Furthermore, the student must know how to turn this unnecessarily complicated thirteen term form into the more readily understandable two term form. 5. The Expansive Approach In this approach a much wider look at the topic of second order differential equations is taken. A range of the facilities of the computer algebra system (not just its manipulatory ones) can be used to explore the key mathematical concepts and to interpret (not just obtain) the solution of the problem. Knowing that the computer algebra package will save the time previously required to explain the algorithm and for repetitive practice other elements can be emphasised. More time can be spent on the derivation of the equation, for example, as a model of a mass-spring-damper system. The linearity of the general differential operator of this kind can be explored using a variety of simple and more complicated functions. Part of such an investigation is shown in the worksheet in Figure 4 below.
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y1:=exp(-t): y2:=sin(t): L:=diff(y(t),t,t)+a*diff(y(t),t)+b*y(t);
L1:=simplify(subs(y(t)=y1,L); L2:=simplify(subs(y(t)=y2,L));
Lsum:=simplify(subs(y(t)=y1+y2,L)); L1+L2-Lsum;
Lprod:=simplify(subs(y(t)=y1*y2,L)); simplify(L1*L2-Lprod);
Figure 4: MAPLE worksheet investigating linearity The importance of linearity to the structure of the solution can be easily illustrated on specific examples such as y ′′ + 5y ′ + 4 y = 10 cos( 2 t ) shown in Figure 5. This builds on the understanding of linearity gained from the previous worksheet (Figure 4). L:=diff(y(t),t,t)+5*diff(y(t),t)+4*y(t): y1:=exp(-t): y2:=exp(-4*t): yp:=sin(2*t): L1:=simplify(subs(y(t)=y1,L)); L2:=simplify(subs(y(t)=y2,L));
Lcf:=simplify(subs(y(t)=A*y1+B*y2,L));
Lyp:=simplify(subs(y(t)=yp,L));
Ly:=simplify(subs(y(t)=yp+A*y1+B*y2,L));
Figure 5: Building up the solution
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The effect of a change to the model on the structure of the problem can be explored. For example, students can explore what happens if the damping force is proportional to the square of the velocity rather than just to the velocity. The small investigation shown in the worksheet of Figure 6 illustrates that the equation is no longer linear.
L:=diff(y(t),t,t)+a*diff(y(t),t)^2+b*y(t);
y1:=exp(-t): y2:=sin(t): L1:=simplify(subs(y(t)=y1,L)); L2:=simplify(subs(y(t)=y2,L));
Lsum:=simplify(subs(y(t)=y1+y2,L));
L1+L2-Lsum;
Figure 6: Non-linear damping The graphical features of a computer algebra package can be used to see what the solution looks like. By varying the damping parameter the three kinds of solution (under-damped, critically damped and over-damped) can be explored. The worksheet shown in Figure 7 illustrates such an exploration. A little MAPLE syntax is needed here, but students could avoid having to know this by using more commands. Here the precise mathematical form of the solution of the differential equation need not even be displayed. For this investigation the nature of the solution (oscillatory or non-oscillatory) is more important. Students can explore the relationship between the model parameters at the point of transition from oscillatory to non-oscillatory solutions. The graphical form of the different solutions can also be used to illustrate the idea of a steady state solution for inhomogeneous problems. By then looking at the mathematical form of the solution it is easy to determine the steady state and transient parts. More creative use of the graphic facilities of computer algebra packages to produce animations of the physical situation (see Rozsasi [5]) are also possible.
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de:=diff(y(t),t,t)+a*diff(y(t),t)+4*y(t)=0: ics:=y(0)=0:D(y)(0)=0: y1:=op(2,dsolve({subs(a=1,de),ics},y(t))): y4:=op(2,dsolve({subs(a=4,de),ics},y(t))): y10:=op(2,dsolve({subs(a=10,de),ics},y(t))): plot({y1,y4,y10},t=0..10);
Figure 7: Different Strength Damping The expansive approach presents a much broader picture of the topic, allowing it to be set more firmly in an engineering context. The student is also given more chance to learn by guided discovery. Being able to find the solution of problems with specific values assigned to the parameters is no longer the key learning outcome. Although, of course, the student should learn how to do that using the computer algebra package. In principle much of the expansive approach could be followed without a computer algebra system. However, the time required to do this would be prohibitive. Furthermore, carefully constructed illustrative examples may lose their forcefulness when algebraic errors have obscured the point to be illustrated. It is only the power and accuracy of a computer algebra package which makes such an expansive approach viable. 6. Conclusions It is the author’s firm belief that before long computer algebra will have a major impact upon the teaching of engineering mathematics. It seems likely that in most courses either the minimalist approach or the expansive approach will have been adopted. If all that is wanted is for students to be able to obtain the solutions of given classes of problems then the minimalist approach is probably the right one to take. However, if we want students to have some understand of wider mathematical principles and some experience of moving from engineering to mathematics and back again then the expansive approach is surely a better way forward.
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References 1. Stern, L A, ‘Computer Calculus’, Science News, 1981, Vol 119. Reprinted in SIGSAM Bulletin, 1981, Vol 15, Part 3, pp 26-27. 2. Calmet, J, ‘The role of computer-based symbolic manipulation packages in mathematics teaching for engineers’, Int J Math Educ Sci Technol, 1987, Vol 18, Part 5, pp 663-680. 3. Mustoe, L, ‘Engineering Mathematics’, Addison-Wesley, Wokingham , England, 1997. 4. Attenborough, M, ‘Engineering Mathematics Exposed’, McGraw-Hill, London, 1994. 5. Rozsasi, R, ‘The use of Mathematica programmed virtual experiment animations’, in ‘Mathematical Education of Engineers’,Eds: S.Hibberd and L.Mustoe, IMA Publications, 1997.
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