rlopez

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12 years, 171 days

Dr. Robert J. Lopez, Emeritus Professor of Mathematics at the Rose-Hulman Institute of Technology in Terre Haute, Indiana, USA, is an award winning educator in mathematics and is the author of several books including Advanced Engineering Mathematics (Addison-Wesley 2001). For over two decades, Dr. Lopez has also been a visionary figure in the introduction of Maplesoft technology into undergraduate education. Dr. Lopez earned his Ph.D. in mathematics from Purdue University, his MS from the University of Missouri - Rolla, and his BA from Marist College. He has held academic appointments at Rose-Hulman (1985-2003), Memorial University of Newfoundland (1973-1985), and the University of Nebraska - Lincoln (1970-1973). His publication and research history includes manuscripts and papers in a variety of pure and applied mathematics topics. He has received numerous awards for outstanding scholarship and teaching.

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These are answers submitted by rlopez

The coefficient of f'' is zero at t=0. Hence, a series solution at t=0 cannot be generated from the DE. That is the meaning of the error message. It's not a Maple issue, it's a mathematical one in the DE itself. The fundamental existence and uniqueness theorem for an ODE is based on an equation that can be solved for the highest derivative, and this derivative must be defined at the initial point. If the DE bacomes 0*f''(0)=..., then f''(0) is not defined, and a solution probably does not exist.

Unless the Tutor provides for returning some result to the calling worksheet, it is not possible in general to insert the result of a Tutor experience into the worksheet. At best, you can take a screen-shot and paste that into the worksheet, but of course, that is then a static image. Of course, I would imagine that all users of Maple's Tutors would be happy to see the day when the Tutors are an integral part of the worksheet and not just a pop-up.

The siderels (side relations) option in simplify might be of help.

First, include in the dsolve command the option output=listprocedure. This creates a list of procedures that include R(theta)=proc(theta)... and mu(theta)=proc...

Access the procedures that generate the values of R and mu as follows. Use names other than R and mu. For example:

RR:=eval(R(theta),F);

M:=eval(mu(theta),F);

Then, write just Int(... using RR and M in place of R and mu. Apply evalf to this unevaluated integral. When i did this, I immediately obtained 46087.3499759295 as the value of the integral.

In other words, assigning the result of the dsolve command to F as was originally done does not access the solutions for R and mu in any way useful for computing. It is only useful in odeplot, the command to graph numeric solutions.

Then, writing Int(... = evalf(int(... isn't optimal. Use of Int provides the unevaluated integral. Use of int causes Maple to try to obtain an exact solution, after which, (when that fails) Maple tries to implement a numeric solution. That's wasteful. Since the integration must be carried out numerically, write just Int, the unevaluated integral, and apply evalf to that.

Finally, the use of eval is a handy strategy in Maple when trying to access the left-hand side of an equation. Suppose the equation is x=5 and it is included in a long list L of other such equations. To extract the value of x and assign it to a different name, use X:=eval(x,L) which evaluates the name x with the equation in the list L, thereby picking out the value 5 and assigning it to the new name X.

As Carl Love points out, code can be in plain sight, or it can be in a Start-Up region so that it executes just after the worksheet is launched. There is also the Code Edit region in which code can be inserted, and the region shrunk to an icon. Finally, code can be hidden behind an output in a table cell. Use the Table Propery dialog to expose the code. I tuck the code for graphs behind such table cells so that only the graph itself shows.

Some of the newer worksheets crafted by Samir Khan use even more sophisticated techniques that were explained in a post to this forum within the last year.

normal(expr,expanded) returned B for me. This command simplifies fractions, and the option expands the denominator.

solve(identity(f,x),[a,b]) returns the values of a and b.

map(coeff,f,sin(2*x)) and map(coeff,f,cos(2*x)) returns the two equations in a and b.

Both the Maple worksheet and the document can be used to capture the kind of work you describe. The worksheet, with no typeset math, might export satisfactorily to LaTeX, but the document, less so. Maple's best exports are to pdf or html. The DocumentTools package is designed more for the interaction with embedded components.

Guessing at what your function  might actually be, I tried the following.

f:=piecewise(x<2,x-1,x=2,2,x>2,2-x);

plot(f,x=0..3,discont=true);

The single point (2,2) is graphed above the break in the linear segments.

The example given is an explicit function of two variables: z = z(x,y). Why  not just apply the solve command to the two equations obtained by differentiating with respect to x and y? There are Maple commands for such problems, but why not just take a simple approach?

The "also" is not clear. Is it a question about solving a constrained optimization problem in two variables? If so, I suggest the LagrangeMultipliers command in the Student MultivariateCalculus package. Of course, the Lagrange multiplier method could easily be implemented from first principles, of via the task template in Tools/Tasks/Browse: Calculus-Multivariate/Optimization.

Regression analysis is only available in Maple's Statistics package. See ?Statistics,Regression,Solution for the help pages where this material is detailed.

Functions of matrices are obtained by computations involving eigenvalues and eigenvectors of the matrix. It isn't likely that one will know the exact eigenpairs of a large matrix. I'm not sure what Maple's MatrixFunction command does with a matrix containing at least one float so that the eigenpair calculation switches to numeric algorithms. The OP is welcome to experiment. Put a decimal point after any one of the entries in the 9x9 matrix and see what happens.

The help page for pdsolve,numeric indicates that Maple can solve (numerically) a "single or set or list of time-dependent partial differential equations in two independent variables", What is deceptive is that one of the two independent variables must be the evolution variable, namely, time. That leaves just one spatial variable. The pde provided by the OP has two spatial variables. Hence, it is outside the scope of what Maple can solve.

If I understand correctly:

y:=x^3;

plot([y,diff(y,x,x),x=0..1])

 

mtaylor(P,[x,y],3) - mtaylor(P,[x,y],2)

The first call to mtaylor creates a polynomial whose terms are of degree 2 or less; the second, a polynomial of degree 1 or less. The difference should be a polynomial of exactly degree 2.

I'm sure there is a more elegant way to do this, but I think my approach works works.

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