@shashi598 

First of all, the question:
What is the aim of your work? Is it to solve a numerically specific problem or is it to design a solution algorithm for a category of equations?
In contrast to ordinary differential equations, there is no general existence or uniqueness statement for solutions to partial differential equations. There are solution methods for the classical equations of theoretical physics, but the general theory of PDEs is currently treated individually for categories of equations. And there is a mountain of literature on this, which quickly leads one to sink into the depths of functional analysis.
According to this, two methods have prevailed in practice: the FEM method, according to which the area of ​​the definition domain is discretized/networked and solution ordinates are calculated over the grid points using clever methods from the difference calculation (there is free software for this on the internet) and the minimum method already described. For the latter, you can find information in Michlin's book "Differential equations in mathematical physics", page 81, under "energy functional". In any case, the spectral method you have chosen currently appears to have "only" theoretical value. But the literature describes that in this method the Chebyshev polynomials are numerically more favorable than Legendre polynomials.
As a Maple newbie, I cannot say anything about Maple's capabilities in pde matters.

@Rouben Rostamian  

Yes, this is a well-known graphical solution "for the road" of Pothenot's problem.

As a Maple beginner, I played around with the ODE for practice and chose coefficients. I have difficulty entering initial values ​​(x must logically be different from zero). And I am also unable to numerically approximate the tapeworm terms in the solution. I am asking for help with this in the attached file.

Download test.mwtest.mw

@nm 

It's a must to have fun - but now to the point:
The ODE is a homogeneous linear ODE with non-constant coefficients. Equations of this type are covered in "Kamke, Differential Equations, Solution Methods and Solutions, Volume 1". A similar example is given in this book under 4.37 on page 534 with a solution. And as is to be expected with linear ODEs, the solution is formed from a linear combination with coefficients that are to be calculated from initial values. My first answer referred to these coefficients. In the present case, it is therefore to be expected that the summands contain Bessel functions, for example. And from this point on, the fun of deriving a closed solution as a term stops for me. Therefore, the equation should be converted into a first-order system and treated numerically. This requires the specific specification of the coefficients C_i and the initial values.

Calculating the coefficients requires initial values. Otherwise, only a general solution to the ODE can be determined. First, however, the equation should be divided by c1. Then four initial values ​​are sufficient.

How do you want to calculate the general solution?

@vv 

For me as a beginner, it is impressive what is possible. Yes, Maple seems to be weak when it comes to Diophantine equations. And since it has been proven that there is no general solution algorithm (Hilbert No. 10), categories of equation types should be implemented with their solution methods.

There are methods for tasks of the current type, e.g. https://www.alpertron.com.ar/QUAD.HTM. It even provides recursions that are based on the known basic solutions (3;1) and (5;1).

Unfortunately, I have no idea about IT, I am just a theoretician and that is why I am asking questions here. At the moment I am trying in vain to insert text from another file into a Maple worksheet using copy and paste between Maple lines. I would like some advice.

@Mariusz Iwaniuk 

Thank you, even the Binet form of the formation law is shown. Only the recursions are still missing.

Similar to how sqrt(5) plays a role in the Fibonacci-Numbers, the importance of sqrt(30) becomes clear here. The "30" can be explained if the original equation is transformed to
(x-6*y)^2 + (4*x +12*y - 3) = 30*y^2.
:-)

@Mariusz Iwaniuk 

Your text is difficult to read, it is too small. As far as I can read, the solution (x; y) = (5; 1) is missing. Can recursions be created for the other solutions in the table?

The system of ordinary differential equations to be investigated is linear, homogeneous and has constant coefficients. The solution of such simple systems is described in detail in the literature (Kamke, Pontrjagin, ...). What is being sought? There is no information on initial values. Is only the general solution to be determined or are there specific numerical specifications for the coefficients?

If there are specific initial values ​​and coefficients, the solution of the ODE system in question is very simple with the help of paper and pen after calculating the zeros of the characteristic polynomial. This polynomial is a biquadratic equation.

@nm 

Actually, I asked if the determination of the center of the matrix ring in particular is implemented in Maple. It seems that this is not the case, so we can end our discussion.

@vv 

In the future I have some plans regarding symbolic calculation. I would like to find out where the limits are. As before, I have chosen tasks for this on my journey of discovery.

@nm 

Apparently I expressed myself imprecisely, sorry. The background to my question is the task:
Determine all matrices X that are commutable with every regular matrix with respect to matrix multiplication, i.e. A*X = X*A for all A. This does not exclude the possibility of random hits for X, for which the factors X and A are also commutable.

@Carl Love 

This is a strong solution. But I need a little more time to understand it. The brevity of the solution is impressive. But what happens in the "background" after calling the commands is invisible. A long time ago I solved this task in individual steps using the good old derive. This resulted in two solutions for the auxiliary values ​​(s; x), both of which lead to y=11. But these values ​​are not what the task is looking for.

Thank you very much :-).

@vv 

You have proved a theorem under more general conditions, from which Völler's theorem follows as a special case - congratulations. Please forgive my late comment. I wanted to look into it thoroughly first.

@vv 

Your work shows the imprecise formulation of Völler's theorem and its proof in the original work from 1858. Only when working through his proof can it be seen that important prerequisite for the proof are not mentioned or are only mentioned in passing/superficially. Today the theorem would be formulated differently.

Interpretation in the sense of the original proof (loosely formulated):

In the Euclidean plane, the convex or concave curve section of a four-fold continuously differentiable function f(x) and the points A and B on it as well as the tangents at these points with their intersection point C are given. F is the area of ​​the triangle ABC and G is the area of ​​the curve segment over the chord of length s = AB. The ratio G/F = 2/3 is to be proven for the limiting case B ---> A, i.e. s ---> 0. The function f(x) is assumed to be convex/concave positive with non-vanishing 1st and 2nd derivatives at the points A and B.

Therefore, to explain:

In the original work according to the source, it is tacitly assumed that with a general function approach f(x), the first and second derivatives do not disappear at points A and B and that f(x) is continuously differentiable up to the fourth derivative, so that the Bernoulli/L'Hospital rule can be applied according to the proof procedure. This can only be seen in the course of the proof, including in the sketch and at the end of the calculation. Non-degenerate circles of curvature are assumed at curve points A and B and local extrema are to be excluded there. The sketch used in the original proof shows this without comment, because this is the only way to obtain the non-degenerate position of the tangent intersection point C.

Under these conditions, it is sufficient to evaluate the 4th derivative.