@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.

@C_R Thanks for the data.  I'll see what I can do.

@vv That's a very clever construction. I see that the deviation of the answer from a rational in this case is of the order of exp(-10^28)  which is too small to be detected by fsolve().

@Mariusz Iwaniuk Thanks for this demo which clearly shows the trend.  If find Ronan's explanation is more insightful.

@Ronan Yes, that's it.  Now that you have pointed this out, it looks so obvious to me.  I have converted your "Reply" to "Answer" in order to give it a Vote Up.