@vv That's an interesting observation. I had not noticed that. But now that you have pointed out, I see that the polynomial is actually symmetric about t=1. After replacing t by t+1 it simplifies to a cubic in t^2:

9*t^6 - 87*t^4 + 51*t^2 - 5

@vv Yes, that's the correct solution. The answer may be slightly simplified by combining the sine and cosine terms. I don't know how to get Maple to do that. I calculated the simplified version by hand:

There is an interesting symmetry-breaking in this problem. Since the values of the two volumes are symmetric about α = π, one would think that the maximum will occur at α = π, but that's not so as we see in the solution presented. In fact, α = π corresponds to a local minimum!

@JAMET I made that animation based on what you had proposed, but come to think of it, it's likely that a practical motorcycle engine will be configured to distribute power symmetrically over one revolution, as in this modified version.

The power distribution is still not quite symmetric, but it can be made symmetric by adding two more pistons, making it a 4-cylinder engine.

piston-crank-animation-alt.mw

@nm In my earlier reply I wrote: "The format of the font specification is defined in ?plot,options under the "font" entry. ".  Did you look?

A font specification is expected to be a list of the form [A,B,C] where A is the font family (e.g., Helvetica), B is the fine style (e.g., Bold), and C is the font size (e.g., 18).  B or C may be omitted, but not A. Saying axisfont=[12,12] or axisfont=12 makes no sense.

@Kevin Dragnet As Carl noted, the use of dchange in this context is an overkill, but for whatever it's worth, here it is:

restart;
sys := { diff(x(t),t) = y(t), diff(y(t),t) = -x(t) };
Tr := {x(t) = r(t)*cos(theta(t)), y(t)=r(t)*sin(theta(t))};
PDEtools:-dchange(Tr, sys, {r(t), theta(t)});
solve(%, {diff(r(t),t), diff(theta(t),t)});

The line that defines Tr in the code above may be obtained alternatively through the method that Carl proposed, as in:

Tr := [x(t), y(t)] =~ changecoords([x(t),y(t)], [x(t),y(t)], polar, [r(t), theta(t)]);

but in my view that's also an overkill.

Simultaneous access to multiple help pages will be a welcome enhancement. But I would rather have the multiple pages displayed as tabbed panels within a single help window (as it's done in web browsers and maple worksheets) rather than multiple help windows cluttering my screen.

@mmcdara That's a clever idea!

@Pets71 For the record, I am running Maple 2021 on Ubuntu 20.04 LTS installed on a 10 year old laptop with 4GB of memory. I use it just about every day in my classes for teaching, without a single problem.

I agree with Thomas Richard that what you have described smells very much like a bad RAM in your laptop. That's a hardware problem and has nothing to do with Maple.

See https://linuxhint.com/run_memtest_ubuntu/ on how to test your RAM in Ubuntu.

Isn't it more natural to name your variables S[i,j] rather than Sij?  If you do the former, then you may do

local S;
seq(seq(assign(S[i,j]=Vector(datatype=float)),i=1..9),j=1..9);

It will be good if you showed some attempt toward solving this homework problem.  You don't want someone else to do your homework for you, do you?

@rlopez My original example required a symbolic solution (x(p) and y(p)). Should a symbolic solution be unattainable, we may resort to a numerical solution, which turns out to be even simpler than the symbolic, as illustrated in the following worksheet.

Download area-under-envelope-numeric.mw

@mmcdara As rlopez has noted, the method of lines (MoL) is unrelated to the method of characteristics. When a PDE has both space and time variables, in MoL we discretize space through finite differences, while treating the time variable as continuous.  Thus, the PDE is approximated by a system of ODEs.

About a year ago, I illustrated the MoL in the post https://www.mapleprimes.com/posts/212624-Solving-Multispan-Euler-Beams-With by applying it to the equation of the deflection of a multispan Euler beam.

A wise man once said: "If you don't know how to solve a problem, there is a simpler problem that you don't know how to solve. Try that one first".

In your case, the simpler problem would be the one-dimensional case of the heat equation. Do you know how to do that one?

@vv That's very clever.  Vote up!

Your homework asks you to define a function and then find the optimal solution.  But optimal solution to what? Perhaps you have abbreviated the question too much to be comprehensible.

You attempt to solve f(x,a,b,c)=0 for x.  The x that you would obtain there is called the root of f. In what sense is that related to the optimum of anything?

I can guess what the question is about but that would be only a guess and I don't want to put words in your mouth.  It will be good if you formulate your question in a clearer way.