@janhardo The expression piecewise(f(x) < 0, 0, f(x)) is essentially equivalent to both of

1. max(f(x), 0)
2. if f(x)::realcons and is(f(x) < 0) then 0 else f(x) end if where f(x)::realcons is a relation that evaluates true iff f(x) is a real constant (i.e., contains no free symbols such as 'x').

Of course, 1 would be the preferable formulation, but symbolic integration doesn't handle it.

@vv I didn't read the paper. Is it not exactly true that we expect 1/2 of runs to be length 1, 1/4 length 2, etc. (assuming underlying null-hypothesis distribution is binomial with p=1/2)? Or do you mean that chi^2 does not give exact p-values for discrete cases? I totally agree with that.

Assuming that the underlying binomial has proportion p (1/2 or otherwise), my off-the-top-of-my-head computation is that the distribution of run lengths is

P(run length = k) = p^k*(1-p) + (1-p)^k*p. Do I need to make a finiteness correction to that?

Do you mean that you want to duplicate in Maple the free-body diagram shown in your notes?

@janhardo You asked for a way to check (or verify) your results for 1c. Maple's purely numeric integration is very robust. Do this:

evalf(Int(exp(-x^2)*(x^4 - x^2), x= -3..3));

By using the capital-I Int with evalf, you are requesting purely numeric integration (aka quadrature). The algorithms used for this are completely different than if you had used evalf(int(...)). Since the two answers agree, they are almost certainly correct.

@janhardo P:-C is just another way of saying "command C from package P." It's an alternative to with(P,C) that I strongly prefer.

Regarding f-g: Consider this: "The area between f and g is the area below f (and g) minus the area below g." Does that make sense?

My guess is that if it can be done in Maple, then it'd be done with the "Stochastic Processes" tools in the Finance package. Have a look at the subsection "Stochastic Processes" of help page ?Finance. I don't expect that you'll find a stock solution for your equation, but you may be able to cobble togther something from the given tools.

A Google search on run lengths in random sequences yields some useful results. My first hit was a 2015 paper "New statistical randomness tests based on length of runs." I'd guess that one important application of this is forensic accounting. 

@Preben Alsholm You're more than welcome. Let me know if you want any further definitions. Here's an important one that I should've included. I didn't because it didn't pertain directly to this Question, but it's a very important class of graphs with many practical applications:

Definition 6: A graph with chromatic number 1 or 2 is called bipartite. Of course, the case where the chromatic number is 1 only contains the trivial graphs with no edges.

I find that writing formal definitions is very helpful for clarifying my thoughts. Some of the effort in mathematics education should be redirected from writing proofs to writing definitions.

@janhardo The area between curves f and g is evalf(Int(abs(f-g), x= 1..15)), regardless of which is on top. So, you don't need the intersections. However, they're not difficult to find: There's 1 and all points where sin(x)=cos(x)  => tan(x) = 1. So Pi/4+k*Pi.

@Preben Alsholm Preben, these formal definitions should provide everything that you need to know for this question. I only post this because you said that you know next to nothing about graph theory. So, sorry if this is already familiar material.

Definition 1: A partition P of a set S is a set of nonempty pairwise-disjoint sets whose union is S. The members of P are called the blocks of the partition.

Definition 2: A coloring of a graph is a partition of its vertices such that no two vertices in the same block share an edge. If the coloring has k ​​​blocks, it's a k-coloring.

Definition 3: The chromatic number of a graph G is the minimal ​​​​​​k such that G has a k-coloring.

Definition 4: A k-clique of a graph is a k-subset of its vertices such that every pair of those vertices share an edge.

Definition 5: The clique number of a graph G is the maximal k such that G has a k-clique.

Obvious Theorem: For any graph G, CliqueNumber(G) <= ChromaticNumber(G).

@AHSAN Don't repost this same Question in other threads. And stop using the title "ODE solution".

@Carl Love Alas, specifying gridlines= false no longer works as a workaround for uploaded worksheets.

If you post some of those exercises, preferably with your solution attempts, people here will surely be able to respond.

@dantopa But note that you get the correct results with the 'col' option, as VV showed. When you examine 'col' after running the command, you'll get the color assignments, and you can easily verify that it's a valid coloring. What's more difficult to verify (indeed NP-complete difficult) is whether it's a minimal coloring. 

In Maple's 2D Input, you need to put a space after Pi. Without the space, there is no implied multiplication.