Fundamentals and Modern Technology Working Together.

While there are several excellent computer algebra systems on the market, which can greatly enhance understanding in some cases, this study showed some of the pitfalls of becoming overly confident in the solutions produced by these types of systems. The output from these software systems was convincing and there was a tendency to take it at face value and not question its validity. Students must be taught about the limitations of computer algebra systems. They must be well grounded in the fundamentals so they are able to use them to question the results of these systems. The article also emphasizes that when possible it is wise to check computer algebra solutions using alternative methods. These systems can be used to amplify and help explain some of the fundamentals. At the same time, revealing limitations of these systems to students often serves as a valuable teaching method, which can lead to a deeper mathematical understanding. The substance of this study was the fundamental principle of existence-unique ness as applied to the solution of differential equations.

The fundamental theorem for differential equations answers the question; does a solution exist and is it unique? This theorem is reassuring when it can be applied. As Calculus students learned in Integral Calculus because of the Rieman sums, mathematicians often knew that the integral existed though they did not know how to perform the integration. Similar ideas come to life here because the solution for a differential equation always involves integration. In this study the existence-uniqueness theorems will help in checking the results from Mathematica's differential equation solver, DSolve and Mathematica were used to show more clearly how this theorem can be applied.

The existence-uniqueness theorem along with its proof for differential equations is given in many differential equations books, Lomen, and Love-lock (1999) and Borrelli, and Coleman (1998). Because the theorem will be used to show a solution from Mathematica's DSolve is incorrect, a brief statement of the theorem will be made.

Existence-Uniqueness Theorem

Consider the following differential equation with its related initial value.

dy/dx = f(x, y), y([x.sub.0]) = [y.sub.0] (1)

In this equation [x.sub.0] and [y.sub.0] are the initial values. This theorem indicates that if f(x, y) and [partial]f(x, y)/[partial]y are both defined and continuous in a rectangular region containing the initial values in its interior, then equation (1)

has a unique solution passing through the point y([x.sub.0]) = [y.sub.0]. The solution is defined for all values of x having values inside the rectangular region.

Given this as a fundamental working theorem to monitor the output from a computer algebra system, consider the following example.

Differential Equation 1

Given the following equation:

dy/dx = 3[square root]y (2)

In this example the entities needed to apply the theorem of existence-uniqueness are as follows:

Fundamentals and Modern Technology Working Together

f(x, y) = 3[square root][y] (3)

[partial][f]/[partial][y] = 3/2[square root][y] (4)

It is necessary to know the initial values of interest before the conditional equations (3) and (4) can be tested to determine if the existence-uniqueness theorem is applicable. Equations (3) and (4) indicate that the values of y must be greater than zero. Three different sets of initial conditions will be investigated for this example.

Initial Condition 1.

([x.sub.0], [y.sub.0]) = (0, -1) (5)

Using DSolve to solve equation (2) with the initial condition specified in (5) will not find solutions in the real number plane. When y takes on the value of -1, the two equations (3) and (4) are no longer defined in the real number plane. This example illustrates how closely coupled the differential equation is with the initial conditions specified. The existence-uniqueness theorem tells nothing about the solution for this differential equation with this initial condition. DSolve gives two complex solutions for this initial condition. Both satisfy equations (2) and (5). Because the requirements have not been met in this case, the existence-uniqueness theorem cannot be applied to this initial value problem. Therefore, both solutions are ok. The solutions returned from Mathematica's DSolve are:

[y.sub.p] = (1/4) [(2i-3x).sup.2], and [y.sub.p] = (1/4)[(2i+3x).sup.2]. Consider a new set of initial conditions for the same differential equation.

Initial Condition 2.

([x.sub.0],[y.sub.0]=(0, 0) (6)

Consider this initial condition along with the requirements for applying the existence-uniqueness theorem. There is no problem with condition equation (3). However, condition equation (4) is undefined at the initial condition point. Therefore, the required rectangular box containing the initial condition point in which the condition equations (3) and (4) are defined and continuous cannot be constructed. This means the existence-uniqueness theorem cannot be applied to this system. By inspection y = 0 is a solution of the differential equation (2). Its graph coincides with the horizontal x-axis. This is the equilibrium solution, and it goes through the initial value point (0,0). However, the uniqueness part of the theorem that says that no other solution could go through this point cannot be applied.

Using Mathematica's built-in differential equation solver DSolve, and specifying the initial condition (6) gives the following solution.

Y(x) = 9[x.sup.2]/4 (7)

This is a solution that passes through the initial value point (0,0), as does the equilibrium solution. DSolve does not give any mention of the equilibrium solution, that is y = 0. However, NDSolve, Mathematica's numerical differential equation solver gives an interpolating polynomial that is the equilibrium solution. Figure 1 shows both of these solutions along with the initial point.

In this case Mathematica's DSolve gave a correct solution, but it did not give any clue about the equilibrium solution. It is also a valid solution. Also DSolve did not give any information about why two solutions are possible for the same initial condition. But because fundamental knowledge about the theorem of existence-uniqueness has been applied, more than one solution at the same point are allowed when the requirements for the existence-uniqueness theorem are not met.

Initial Condition 3.

([x.sub.0], [y.sub.0]) = (0,1) (8)

Both [square root] 2 3 / 3 and [partial]f / [partial]y = 3/(2[square root]y) are defined and continuous for initial condition (8). Therefore the requirements for applying the existence-uniqueness theorem are met. A rectangle of convergence can be found in which there will be one and only one solution that passes through the initial value (0,1). The dimensions of the rectangle must be such that the initial condition point is included. Also y [greater than] 0 is important because y = 0 or y [less than] 0 will cause problems for conditional equations (3) and (4). The following box meets these criteria.

{(x, y):x [epsilon] (-3/4,1/3), y [epsilon] (1/4,2)} (9)

Inspection of differential equation (2) gives the following implicit general solution indicated by a subscript g:

2[square root][y.sub.g] = 3x + C (10)

Using the initial condition that y(0) = 1 , gives C = 2 for the constant of integration. This leads to the particular solution indicated by a subscript p:

[y.sub.p](x) = 1/4[(3x+2).sup.2] (11)

Figure 2 shows a plot of this solution, the bounding rectangle containing the initial value, and the initial value. Figure 3 verifies the solution by plotting it superimposed upon the slope-field for the differential equation (2). The solution indicates no problem with the dimensions of the original bounding rectangle given in equation (9). Because equation (11) is defined for all values of the variables, the bounding rectangle could be enlarged.

DSolve gives two solutions as output for this differential equation (2) and the initial condition equation (8). They are labeled (12) and (13).

[y.sub.p](x) = 1/4[(2-3x).sup.2] (12)

[y.sub.p](x) = 1/4[(2+3x).sup.2] (13)

Equation (13) is recognized as the same result found by inspection and listed as equation (11). However, equation (12) is an entirely new solution. In this example NDSolve gives an interpolating polynomial corresponding to the correct solution (13). Figure 4 shows both these solutions plotted together. Both equation (12) and (13) pass through the initial condition point shown in equation (8). Because the differential equation and its initial condition satisfy the requirements of the existence-uniqueness theorem, this should not happen. Plotting both solutions in Figure 5 superimposed against the slope-field for differential equation (2) shows that solution (12) given by DSolve goes against the slope field. Using Mathematica's PlotVectorField instruction points out clearly that equation (12) is incorrect for this initial condition. Therefore, an alternative approach using a second Mathematica instruction, PlotVectorField proved that DSolve gave a wrong answer for the initial conditions specified.

This problem illustrates how instructors and students must not rely entirely on any computer algebra system. Instructors and students must couple the utilization of these powerful systems with some basic knowledge such as the existence-uniqueness theorem. The existence-uniqueness theorem led the way in identifying the error and the PlotVectorField instruction verified the error. Using the theorem of existences-uniqueness we have discovered an error in the output from DSolve. By considering why this error occurred we can give our students deeper understanding of the application of this theory.

The problem seems to be in the method used by DSolve to evaluate the constant of integration using the initial condition. As shown earlier if the constant of integration is evaluated while the solution is in the implicit form solution (11) or the same as DSolve's solution (13) results. However, transforming the implicit form of the solution equation (10) into explicit form by solving for y before evaluating C there are some complications.

Squaring both sides of equation (10) and solving for y gives the fob lowing general solution:

[y.sub.g](x) =1/4(9[x.sub.2] + 6Cx + [C.sup.2]) (14)

The initial condition (8) can now be used to evaluate the constant of integration.

y(0) = 1 = 1/4(9 * [0.sup.2] + 6C * 0 + [C.sup.2]) = [C.sup.2]/4 (15)

This gives C = [+ or -][square root]4 = [+ or -]2. Substituting the two values into equation (14) and algebraically manipulating gives the two solutions that DSolve found, (12) and (13). The slope-field has made it clear that solution (12) is wrong for the initial condition specified. The derivative of equation (12) should equal 3 times the square root of the right-hand side of equation (12).

d[y.sub.p] (x)/dx = 9/2(x - 2/3) (16)

The right hand side of (16) should be equal to 3[square root][y.sub.p] (x). Rearranging (12) and substituting gives the following:

3[square root][y.sub.p] (x) = 3[square root]9/4[(x - 2/3).sup.2] = 9/2[square root][(x - 2/3).sup.2] = 9/2\(x - 2/3)\.

If the term in the absolute value symbol is negative, then it must be written as -(x - 2/3). So, this term depends on the initial value. In this case the initial value of interest is x = 0. For (x - 2/3) to be positive it must be true that x [greater than or equal to] 2/3. This is clearly not the case for the initial value given in equation (8). Therefore, for the DSolve solution (12) the following is true:

d[y.sub.p]/dx [neq] 3[square root][y.sub.p] (17)

Upon substitution (17) becomes (18).

9/2(x-2/3)[neq]-9/2(x-2/3) (18)

Therefore, solution (12) from DSolve is not a solution for the differential equation at the initial value given in equation (8). The existence-uniqueness theorem holds, and there is indeed only one solution in the region of interest. The problem is not that DSolve considers y'(x) = 3[square root]y(x) as two different differential equations where the square root is understood to represent both branches of the square root function. DSolve does not solve two separate differential equations one using the positive square root and the other using the negative square root. DSolve uses Solve to find the value or values for the constant of integration in the general solution using the initial condition. In our example Solve gives two values for the constant of integration. Neither value constitutes a parasitic result for the algebraic solution. Solve can find and reject in most cases the parasitic solution to algebraic equations. An example of this is the equation [square root]2x - 1 + [square root]x = 2. This algebr aic equation has two roots, x = 1 and x = 25. The second solution is a parasitic solution and is rejected by Solve. The solution verifier in Solve rejects this solution. This task is very difficult, and is sometimes ambiguous or impossible even in the Solve function. The analogous task in DSolve is even more difficult. The Technical Support staff at Wolfram Research, Inc. has informed me that, "there are not currently any known, practical, general ways to reject parasitic solutions in DSolve." They go on to say that, "Until an algorithm for this purpose is discovered or invented, the type of behavior that you reported may be changed in selected cases, but it will not be possible to avoid it in general." The detection of parasitic solutions generated in DSolve is an active area of investigation among WRI technical staff

This example emphasizes to teachers and students that there is more to solving differential equations and doing mathematics than learning how to use a computer algebra system. Mathematics instructors and students of mathematics must continually be on the lookout for the little subtleties that make our discipline interesting and exciting. It is important to note that in this case the instructor can use Mathematica to help find the error in Mathematica's DSolve. The plot of the vector field using PlotVectorField shows up the error in the differential equation solver DSolve.

If the same algebraic check is performed on solution (13), which behaves properly when compared to the slope-field, it is found that x [greater than or equal to] - 2/3 is what is needed. This requirement is met by the initial condition given in equation (8). Therefore, that solution would also check out in the original differential equation as a satisfactory solution.

Differential Equation 2

Mathematica is used to supplement the study of the following differential equation.

dy/dx = 3x[y.sup.3], y(3) = 1/2 (19)

For this equation f(x, y) = -3x[y.sup.3] and [f.sub.y](x, y) = -9x[y.sup.2]. Because these functions are continuous for all values of the independent and dependent variable, the existence-uniqueness theorem guarantees a unique solution at the initial value point. An arbitrary bounding rectangle can be chosen with the understanding that the interval allowed by the final solution might require an alteration of the dimensions. The following rectangle containing the initial value (3,1/2) was selected at this stage:

2 [less than] x [less than] 4, and 0 [less than] y [less than] 1 (20)

Separating the variables and solving or using Mathematica's built in solver; DSolve gives the solution for this equation with its initial condition.

y(x) = 1/[square root][3[x.sup.2] - 23 (21)

This solution reveals that the bounding box selected above will need to be altered. The solution in equation (21) shows that x must be greater than [square root][23/3], which is about 2.8. This reduction from the original bounding box to a sub-region can be shown in a single graph constructed with the help of Mathematica. Figure 6 shows the direction field, the initial value, the solution, the first guess at a bounding box, and then finally the smaller sub-region required by the solution. The direction field supports what was expected based on the theorem of existence-uniqueness. This graph drives home the point that the theorem only guarantees a solution on a subinterval of the original interval. Plotting a direction field by hand is computationally intensive. Using PlotVectorField is a good example illustrating how technology can be used to help our students gain a deeper understanding of the concept of existence-uniqueness.

But if the solution was not known how could the reduced size of the box be discovered? If the numerical solver, NDSolve is used on the differential equation in (19) with the interval of integration indicated by the original bounding box size given in (20) it returns the following message:

NDSolve::ndsz : At x = = 2.7688850093549373', step size is effectively zero; singularity suspected.

Within the significance of the computer, this is the value of the independent variable that gives a zero in the denominator of (21). The numerical solver indicates that the initial interval must be shortened as is indicated in the true solution. In addition to the above output from NDSolve an interpolating polynomial is given that is valid in the sub-interval, 2.76889 [less than] x [less than] 4. This capability allows the use of a numerical solver to find the subinterval containing the initial value that is valid for the solution of the equation. This procedure can be used for equations that cannot be solved analytically as will be illustrated in the next example.

Differential Equation 3

dh/dt = 5[t.sup.2][h.sup.2] - t/(2h-3), h(1) = -0.379 (22)

Differential equation (22) cannot be solved analytically. First determine if this equation meets the criteria required by the existence-uniqueness theorem. The right hand side of equation (22) is f(t,h) = 5[t.sup.2][h.sup.2] - t/(2h-3). Provided h [neq] 3/2, any other values will be acceptable. Therefore the initial value given is satisfactory. The partial derivative with respect to 12 of the right hand side of equation (22) is [f.sub.h](t,h) = 10[ht.sup.2] + 2t/[(2h - 3).sup.2]. As in the case of f(t,h) if h [neq] 3/2 the requirements for the application of the existence-uniqueness theorem are met. Because the initial condition stated meets these requirements, the theorem guarantees that there will be a single solution existent in a sub-region containing the initial value.

A bounding rectangle excluding h = 3/2 and containing the initial value stated in equation (22) is as follows:

0 [less than] t [less than] 4, and -2 [less than] h [less than] 5/4

Figure 7 contains the direction field, the initial value, the numerical solution, the first guess at a bounding box and then finally the smaller subregion required by the solution. The direction field supports what the theorem of existence-uniqueness predicts. NDSolve returns an interpolating polynomial that is valid over the sub-region,

0 [less than] t [less than] 2.014.

CONCLUSION

Computer algebra systems are now being developed that can serve as excellent aids to teaching, but instructors must be on the lookout for errors. Often the errors or limitations can be used to teach the importance of mathematical fundamentals and thereby lead to a deeper mathematical understanding such as the theorem of existence-uniqueness stressed in this study. This study emphasizes that math instructors must continue to teach the fundamental theories. Students must be taught to understand and interpret the results produced by computer algebra systems in light of the known mathematical theory. Teachers and students must not allow these systems to do their thinking for them. These systems should be used to help teach the fundamentals. The final two figures were generated using VisualDSolve, a package that can be added to Mathematica's library of packages. This package is described by Schwalbe and Wagon (1996)

Author Note

Charles B. Wakefield is an assistant professor in the Department of Mathematics and Science at the University of Texas of the Permian Basin. His e-mail address is [e-mail]. He will send a 3.5" diskette with the Mathematica instruction set used to produce the figures in this articleto interested parties upon request.

References

Borrelli, R.L., & Coleman, C.L. (1998). Differential equations: A modeling perspective. New York: John Wiley & Sons.

Lomen, D., & Lovelock, D. (1999). Differential equations graphics models data, (pp. 56-63). New York: John Wiley & Sons.

Schwalbe, D., & Wagon, S. (1996). VisualDSolve, Springer-Verlag Publishers, Santa Clara, CA.

Wolfram, S. (1999). The Mathematica Book (4th ed.). Cambridge, MA: Cambridge University Press.

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