@Markiyan Hirnyk   I do not understand what you mean by this example? What sense to seek eigenvalues of a matrix consisting of polynomials?

mattcanderson1  If you are interested in a more detailed analysis of a curve of the second order, then you can download my procedure  QuadricCurveAnalysis  (in Russian)  here. In addition to making Maple for your example, it also finds: the eigenvectors and the eigenvalues of the corresponding quadratic form, the canonical equation of the parabola in the coordinates  x' , y', the parameter of the parabola  p , the equation of the axis of symmetry of the parabola, the connection between the original (in x , y  coordinates) and canonical coordinates, the angle of rotation of the canonical coordinate system relative to the original coordinate system and plots the parabola in original and canonical coordinates.

Here is the result of the work the procedure for your example: 

@Markiyan Hirnyk  I were writing my answer regardless of yours, and just before to post I saw your answer. I think for the questioner it's useful to read both answers because in some details, they are not identical.

@yunlongwang  You can not operate with  arbitrary matrices, only with matrices of predetermined sizes. You will not even can define an arbitrary matrix. See

A:=Matrix(n, symbol=a)  assuming n::posint;

  Error, (in Matrix) dimension parameters are required for this form of initializer

@Rouben Rostamian    Maybe   ode[i]$i=1..3;

@Carl Love  I remember that Maple does not recommend to solve the equations for expressions other than the names. Maybe I'm in something wrong.

solve(x^2+1=0, x^2);

@Markiyan Hirnyk  to restart and get result from the previous assumption.

@Markiyan Hirnyk  This is only a special case. The plane  l*x+m*y+n*z+p = 0  is defined, if  l^2+m^2+n^2<>0 . The condition  l<>0  is not equivalent to it.

@mskalsi  In procedure coefff  elements of the list  T  should be names. I do not see in your file any differential expressions, only indexed names. Therefore, for expression in your file procedure should work. If you have any differential expressions, you can just replace them with names foe example  u[x,x]  for  diff(u(x,y),x,x)  and so on.

@Preben Alsholm  for clarifying. I thought that SCR is a normal screenshot and did not know about this option in "More"

@Markiyan Hirnyk  Thanks for answers. I just do not understand why SCR is needed.

@Christopher2222  you are right. Probably the algorithm is based on finding the equation of the plane by the first three points and then verify that the fourth point lies in this plane. It is clear that this algorithm fails, if the first three points lie on the same straight line as in my and Hirnyk's examples.

@Markiyan Hirnyk  

1) I did not understand - what your example, in principle, different from mine?

2) I do not know what  SCR  means.

@Carl Love  Should be  omega(m)  instead of  omega(n)

@Marcel123  

1) I do not know the cause of the error in your calculations. I have no errors occurred.

2) In fact, the results are very similar . If you add to the code two options   abserr=10^(-16), relerr=10^(-16) , the results fully coincide.

Addition.  I think I understood the reason for the error, which you write. If you resolve your differential equation, you must remove the previous assignment for  x , for example  x:='x' . But I do not understand why you are re-solve the equation. You can reuse already found the solution  sol , to find the values of the function  x  and its derivatives for different values  t  and to plot their graphs, for example:

x(0.5),  x(1),  Dx(1), D2x(1.5) ;

plot([x, Dx, D2x], 0..1.5);