Download DataFrames_Select.mw

I won't solve your problem, but I get you started with the Maple problem - Explore command. You are missing the "parameters" parameter. (I also suggest adding the "initialvalues" parameter.")  The trough appears correctly for only certain values, so you might want to check the equations. (I added a bogus area function.) If you don't like the area in the caption, it could be put into the title. Trough.mw

Excuse the jpeg, but Mapleprimes will not display the 2021.1 code. 

 

Download simplify_example.mw

Download Plotting_higher_derivatives_numerically.mw

I think if the conditions are satisfied, it is solvable. I rewrote the integral in the form of constants. 

print(G) is probably the simpliest to show all the values within G. 

@tomleslie's was spot on. May I suggest an slight modification when you use dsolve - use the "output = listprocedure". Then you can extract out the procedure for x(t) and use it like a function. 

For fun, I tried out your toy problem. In the end, I agree with Tom - I don't see the difference and neither does Maple. I used pdsolve, not dsolve. 
 

Download pdsolve.mw

Here is alternative approach that may mimic what you read in the textbooks...

Download Ellipse.mw

Substitute the { } for [ ] and it will be a list of lists.

@wolowizard
And here is the follow up to my terse reply late last night. It adds on to what @Carl Love nicely stated.

If you want to learn more about "lists" vs. "sets" vs. other data types, I recommend chapter 4 of the programming manual. (And one does not require switching out of 2D math to use it.)

https://www.maplesoft.com/support/help/Maple/view.aspx?path=ProgrammingGuide/Chapter04

f''(w) = 0

For a first derivative condition: 

f'(w) = diff_f_at_w_expression

Whether I use an indexed variable vs. an atomic one depends upon the problem I am trying to solve and how I wish to solve it. For example, if I have wish to write or refer to elements in a sequence, list or vector, being able to use indexed variables is quite valuable in readability and worksheet real estate. E.g. seq(a[n]*x^n, n=0..3). Such choices are made when I am writing code, which I perform nearly always within a Code Edit Region.  

However, if I am writing mathematical equations that I want non-Maple users to be able to read, particularly if including a subscript is informative, then I use an “atomic variable”. For writing mathematical equations and functions I use 2D-input because it makes the equations more readable and reduces the content on a page. The output of a 2D-input is repetative.

I agree with Carl Love, the atomic variable leads to fewer problems, particularly with new users, which, clearly, you are not. Many of the students in my courses prefer the image of the subscript provided within 2D input, such as F__x or F__Earth. However, when constructing such variables, too often they end up with the equivalent of F[x], only to discover that should they assign a value to x, their worksheet starts producing unexpected outcomes. Thus, I encourage them to avoid subscripts until they are more familiar with Maple.

There is a post on the evaluation of subscripts in Maple by Robert Lopez that might be of interest to readers: https://www.mapleprimes.com/maplesoftblog/208838-The-Saga-Of-The-Subscript

Personally, I enjoy atomic variables. They truly do make Maple input more readable, i.e., more aligned with what one sees in papers and textbooks.

Use 2D Input. Then, and only then,  the "dot" notation is available and can be found in the Calculus palette. 

When I am in your situaiton, I hit enter after the "x" to add another line. Then I hit the "Remove section" button on the Toolbar. (It is on the right in the circled region shown below.) This ends Section 1 and steps-left the cursor into the "Chapter 1" section. Then I hit "Insert section" button to add another section and step-right. I find this step-left / step-right structure logical since it is akin to what one uses in Word. 

If you end up with more lines in a section then desired, <ctrl>-<delete> (or <command>-<delete> for Mac) works to remove the entire line. 

I could see using this plot in my own courses. Others here are likely to give you some slicker ideas, but here is a start. All the plots are "plot", but one is of "style=point". I put the commands in a procedure so that you could pass it different functions.

 

Download NeatPlot.mw

(Note - the grid lines don't show up when you execute it.) More work using textplot is needed to add the values of f(x) at x.