I use Maple 2026 (and used 2025)  frequently using a VPN and frequently without a VPN without any issues. The only settings I can think of that might be relevant are file->options->network->enable MapleCloud connection, which I leave unchecked, and I am not logged in to my Maple account.

Maple in installed locally on my machine and the licence file is also local.

As far as I know, the only Maple tools for functional equations are in the LinearFunctionalSystems package, but this is only for systems with one independent variable. Perhaps your example can be coerced into this form, or into a system of differential equations, but how to do that isn't immediately obvious to me.

@vv I didn't really expect to get an answer with the OPs example; it was more about how to use gfun. But I was lucky. Your example can be done with a bit more work.

Download gfun2.mw

@acer Thanks. I had tried forget(evalf), forget(`evalf/int`) and gc(). By commenting out, it seems the key one is forget(..,nm), which I would never have guessed.

Actually, I realized I want t as a local in the module, which then means I need H( f, 10000, F:-t ); with kernelopts(opaquemodules = false). Inelegant but it works.

@Alfred_F If you want more, you can use a range. QuadraticIsolve(P, 0 .. 15) gives 62 solutions.

@salim-barzani 

@salim-barzani I updated the answer to include the fsolve commands, using search ranges based on the plot. Each case with different numerical parameters will be different. There are two examples. The solutions are at the intersections of the red and blue curves.

I don't know why it was deleted, but in the past I have deleted ones that are duplicates and instead of telling the person that it was deleted because it was a duplicate they get told it was spam, which is clearly a problem with the system.

At least in Windows, the alt key shows menu shortcuts, e.g., F pops up above the file menu, then alt-F opens the file menu and shows, e.g., O for open and FO for file open. This is common Windows behaviour, but is an alternative to the ctrl- shortcuts.

@MAXR There is a difference between the dots. Anyway, I am currently traveling so perhaps someone else can help you.

@MAXR Please upload your revised version.

@sand15 Some years ago I tried to get my university to renew its site licence, which was very reasonably priced. At the time they were in financial cutback mode, so it wasn't possible. But I did talk to the mathematicians, many of whom had used Maple in the past, but now they have all (but one) changed to SageMath.

A lot of this is driven by cost, and free open source alternatives are now available. Similarly, many engineering departments are changing from matlab to python, again as a cost-saving measure.

So I think the nice interface is an attraction for teaching mathematics and that is a good market. I keep answering questions on stackexchange using Maple, in the hope that is some good advertising. 

You have used a dot "." (matrix multiplication) instead of "*", so you should start by fixing that.

@janhardo I don't understand what you are saying. Are you making some fine distinction between a proof and an algorithm? Many proofs are algorithmic.

@janhardo You could, but the standard way would be via the characteristic polynomial as @Rouben Rostamian used (and presumably Maple's rsolve uses).