@acer The abbreviation of EmpiricalDistribution to Empirical that you've used seems to be undocumented. I don't have any problem with that; I just thought that it should be pointed out. I think that it's ridiculuous and unfortunate that so many names in Maple have redundant words tacked onto their ends. The GraphTheory package is one of the worse offenders.

Please don't repeat Questions from other threads. I read your other Question (the one I just deleted) in it's original thread yesterday, and I don't know the answer.

@emendes Change both instances of degree(p) to degree(p, var). The var at the end of the that line of code does nothing, and can be removed.

@dharr Thanks. I just noticed that myself, and now it's corrected.

@dharr I know that they don't appear in the specific example given, but surely 3*x should be considered linear and y^2 should be considered nonlinear.

@Mariusz Iwaniuk But notice the superscripts on the HypergeometricUs in the final series. They indicate derivatives that Mathematica can't expand. So, that answer is essentially the same as my series(' 'KummerU' '(p, 1/2, t), p= 0, 3).

@Shaaban Do you not understand the relationship between series and derivatives? 

@Shaaban 

KummerU(p, q, t) is (by definition) a solution of the ODE  t y'' + (q - t) y' - p y = 0. The corresponding KummerM is the other solution. The independent variable of the ODE is t, so it's relatively easy to differentiate the solution, y, with respect to t. 

I said earlier that Maple didn't know how to differentiate it with respect to p. That's not 100% true. If you do FunctionAdvisor(KummerU(p,q,t)) and expand the section "differentiation rule", you'll see that there's a fantastically complicated formula of an infinite series of improper integrals and GAMMA functions given as the derivative with respect to the first parameter.

Do you have some reason to want the series with respect to p, or is it just curiosity?

It looks like you're trying to illustrate a linear programming problem. If so, what is the objective function, and do you want to minimize or maximize it?

Just to clarify that for other readers: The OP wants an example of using PDEtools:-PolynomialSolutions(..., HINT= ...).

Is the new larger file size a problem for you, or is it merely a curiosity?

@saketh I suspect that what you want isn't possible. Here's my reasoning by analogy: Suppose that you were computing a power-series solution to an ODE by, say, the Froebenius method. In all but some trivial cases, it would be impossible to compute the degree-n term without first computing all the lower degree terms.

@saketh Thank you. Your example completely illustrates your Question, which I now understand. Unfortunately, I have no answer, but maybe someone else does.

@9009134 I present here the two plots that we're talking about to make it easier for other readers who might be able to help with this. Both are constructed from 605 raw data points, and both have been "normalized" as we discussed above. 

This is "s1".

This is s2:

But could you define "dominant mode" for me? 

It appears that there is a horizontal shift of about 1/2 period between the two plots. Does this need to be taken into account?

Could you give a small example of a PDE which has two polynomial solutions, one of which is returned by PDEtools:-PolynomialSolutions and the other of which is the homogeneous polynomial that you want?