The characteristic polynomial is of the fifth degree:
LinearAlgebra:-CharacteristicPolynomial(m, lambda);
so we don't expect a solution in terms of radicals.  Is there a reason you expect an exact value?

Operating system: Linux

Bug report summary: Maple 2019.2 fails to load the user initialization file   ~/.mapleinit

How to reproduce

1. Put a printing message in the initialization file. For example print("Hello Maple User");
2. Open a new worksheet.  You should see the message "Hello Maple User".
3. Within that worksheet do "restart;" one or more times.  You should see the message "Hello Maple User" printed each time indicating that the initialization file is being read with each restart.

That's all good.

Now instead of opening a new worksheet, open an existing (that is, a previously saved) worksheet. You will see the "Hello Maple User" as expected.  But if within that worksheet you do a restart, you won't see the "Hello Maple User" message.  The initialization is not being read!  Any definitions/settings that you may have in the initialization file are no longer accessible.

@Jjjones98 Okay, then an explicit solution will exist for some but not all choices of f and g.  If you specify f and g, then you can ask Maple to find explicit solutions, if there are any.

@Kitonum Very nice.  Vote up!

@Carl Love The four normals are concurrent means that all four meet at the same point.

I can't make sense of this question.  It appears to be referring to the equation

diff(h(x,y,z),z) = I*(A*f(x,y,z) + B*g(x,y,z));

then it evaluates the equation at z=R where R is some constant.
In Maple's notation that is written as

D[3](h)(x,y,R) = I*(A*f(x,y,R) + B*g(x,y,R));

But since the left-hand side has been evaluated at z=R, we no longer have a differential equation.  You need to be specific in posing your question.

To make this post (and your other posts) useful, I suggest that you begin with stating the purpose of your code.  And at the end give an example of its use.  As is, it is difficult to understand the significance of what you have done.

From the post's title it appears that your code is related to the Euler-Bernouli beam equation.  But that equation is simply d^4 u / dx^4 = f(x),  the solution to which is a matter of integrating f(x) four times and applying the boundary conditions. This can be done in a single line by calling Maple's dsolve function. Does your code provide anything beyond that?

@MapleUser2017 You are doing 
LibraryTools:-Save(Mylib, LibLocation);

That should be

LibraryTools:-Save(MyMat, LibLocation);

Show what you have done so far, and if you have specific questions, post it here to get help.

You don't want someone else to do your entire homework for you, do you?

That's an odd set of differential equations.  Are you sure you are not confusing a differential equation with its solution?

If we take the equations that you have given literally, then asking for a phase portrait is not a good question.  A phase portrait is meaningful concept for an autonomous system, that is, a system of differential equation in which the time variable t does not appear explicitly.  For instance,
x1'(t) = 3*x1(t) + 2*x2(t),
x2'(t) = 5*x1(t) + 1*x2(t),
is an autonomous system, but yours has sin(t) and cos(t), and therefore it isn't autonomous.

@ecterrab Things work fine now.  Thank's for the quick fix!

@ecterrab 

Edgardo, I installed the latest Physics update.  The problem with the `+` is now resolved but `-` needs your attention.

Download mw.mw

@vv Thanks for the suggested workaround.  I would have used it, but with Edgardo's latest fix that won't be necessary.

Edgardo, thank you very much for your lighteningly fast remedy.  I will download and install the update now.

And while we are on the subject of Physics[Vectors[, I have a question for you.

The standard orthonormal basis vectors are _i, _j, _k.  Is it possible to introduce another triple of vectors, let's say _a, _b, _c, and tell Maple that these are also orthonormal?  That is, we want Maple to automatically apply the rules  _a . _a = 1,   _a ._b = 0,   _a &x _b = _c,  etc, whenever these vectors are encountered.

I recall having read reports on MaplePrimes of instances where Maple produces corrupt files when the file name contains Danish characters. I suggest limiting file names to ASCII characters to avoid the issue.