@mon It would be easier if you specified the form that you want to convert to.

The graph of a paraboloid is not bounded: It extends to +infinity. So what do you mean by "I want the full paraboloid to show"?

I don't think that you ask an excessive number of questions. There's no need to be shy.

@aoakindele I have not read the code, so I'm not sure about the finer details of this, but this is my best guess at Maple's numeric BVP algorithm based on many years of experience using it. (And you learn a lot more about the inner workings from the error messages than when it runs smoothly.)

1. The system of ODEs is converted to a first-order system by introducing new dependent variables.

2. The derivatives are replaced by finite-difference approximations, so now it's a system of algebraic equations.

3. A mesh of evenly spaced points is defined on the solution interval. 

4. The algebraic equations are solved at the mesh points by Newton's method.

5. The mesh is doubled. Interpolation is used to obtain a Newton's method initial guess at the new points. Go back to step 4, or quit if sufficient accuracy has been obtained.

From the point of view of the parts of Maple that perform polynomial algebra, such as factor, your expression is identical to

f(4*x) + 2*f(2*x) + 1

for some arbitrary symbol f. This is why it isn't immediately factored.

@Carl Love Sorry, I misread the Question (the most important part being in its title), and this Answer should be ignored. I thought that the goal was to execute a single execution group and that "section" was being used informally to mean "execution group."

This Reply is simply to clear up some confusion about a few of the terms that occur in this and similar threads. 

This is a boundary-layer fluid dynamics problem. I have never formally studied this subject; everything that I know about it comes from answering MaplePrimes Questions since BVPs about these problems are one of the most-common sources of Questions here. (It's a very good bet that any ODE BVP that you see on MaplePrimes with eta as the independent variable is a boundary-layer problem.) Boundary-layer problems consider the dynamics of a system of fluids as if there were distinct masses (or layers) of fluid. There are a vast number of named parameters in these problems. The skin friction is the friction between the fluid layers; it's usually denoted as C__f. The Nusselt number is the ratio of convective to conductive heat transfer at the boundary; it's usually denoted Nu (in Roman, not Greek, letters). The Reynolds number is related to at what point the boundary-layer model breaks down into turbulent mixing of the fluids; it's usually denoted Re.

All hyperlinks above are to Wikipedia articles.

@Madhukesh J K Okay, in that case the Nu vs M plot (for several phi) that you want is the middle plot of the three in Tom's most-recent Reply. This plot could be improved (made smoother) by using a greater number of values of M (Tom only used 3). Other than that, it seems to match the plot that you got with other software.

@nm Yes, I understand all that. The issue is primarily linguistic: You (even still!) continue to incorrectly refer to generalized eigenvectors as eigenvectors, as if the generalized eigenvectors were some special type of eigenvector. It's actually the other way around: Eigenvectors are a special type of generalized eigenvector. So, when you mean "generalized eigenvector", you must always include the word "generalized". The phrase cannot ever be shortened to simply "eigenvector", even if it's in the very next sentence. Doing so will cause confusion. The textbook passage that you quoted does not make this mistake. Read it again carefully. I am not saying that you are misinterpreting the mathematical information in the passage; on the contrary, it appears that you understand well the mathematical distinction between eigenvectors and generalized eigenvectors. I mean that you should reread it to see that they never refer to a generalized eigenvector as simply an eigenvector.

Unfortunately, there is nothing in the language itself that can help you avoid making this type of conceptual error. You just need to know the mathematical definitions involved. In the vast majority of cases, the phrase A B (where A is an adjective and B is a noun) does specify a special type of ​​​​​​B. For example, a black cat is a special type of cat, and if a specific black cat is meant, it can be referred to as simply "the cat" in subsequent sentences without any chance of confusion. I think that exceptions to this rule are more common in mathematics than in other fields. When those exceptions occur, it's often the case that the B is a special type of A B. I suppose that the reason that this arises is that B was defined first and the usage long established before someone decided that their was a need for a super-category of B. That seems to me like a likely explanation for the case of eigenvector and generalized eigenvector. 

@Madhukesh J K The formula that you've given so far requires the Reynolds number (Re) to compute the Nusselt number (Nu). But you have not provided the Reynolds number!! It appears nowhere in your formulation of the problem!! We've been trying to tell you this repeatedly.

@itsme I agree totally with you, and I was about to write something that said essentially the same thing.

@Anthrazit What other software are you familiar with that correctly does what you're asking for? It's not a standard feature of end-user software.

@acer My guess is that the CSV file contains instances of two consecutive commas with no intervening characters other than possibly white space.

@VaughanR Back ticks that enclose no characters denote a symbol, the empty symbol. It's not quite the same thing as NULL, but it works well in matrices because no characters are shown in its display. You could also do subs(0= NULL, A) or subs(0= (), A). These give true NULL values, but are ugly in matrices because they prettyprint as NULL.

The problem with any of these approaches: What if you have a datum that's legitimately 0?

@itsme Your points about Code Edit Regions (CERs) are well made. I never use them for codes less than 50 lines, so my experience has been different than yours. Yes, if I made every execution group in a typical worksheet into a CER, I would find them very difficult to work with.

You are slightly misusing the terms surface, level surface, and solid. An equation (not an algebraic expression) in 3 variables such as x^2 + y^2 + z^2 =1 defines a surface: an essentially 2-dimensional object that lives in 3-dimensional space, such as a sphere. A solid is an essentially 3-dimensional object, such as a ball: a sphere and the volume it encloses. An example of a relation defining a solid is the inequality x^2 + y^2 + z^2 <= 1. An expression (or function) of three variables such as f(x,y,z) = x^2 + y^2 + z^2 has level surfaces, which are the surfaces defined by f(x,y,z) = L for specific levels L.

@a_simsim It is possible to generate the numeric data without plotting, but why do you ask? I find Tom's method quite convenient, and the marginal effort required to produce the plots is insignificant.