@Carl Love As I requested before, please respond to my Answers. Does the above do what you want?

@Bob Sullentrup You wrote:

• What do you mean, "only numerically"? Do you mean there is no algebraic solution...?

Yes: Except for a few simple cases, there is generally no algebraic solution for the inverse of a polynomial of degree 5 or higher. This is a very famous theorem: See Abel-Ruffini theorem.

• ...but the only thing you can do is to  calculate values of the polynomial and then switch x and y?

It's not quite that crude. There are numerous numeric solution techniques (which given a numeric value of y find the approximate numeric value(s) of x) that are more sophisticated than that, such as Newton's method. These methods can usually find inverse values to an arbitrary number of decimal places nearly instantaneously. 

• And what is the function 'fsolve'? Some computer language? Doesn't look like Excel. 

Lol. The name of this forum is MaplePrimes because it's primarily about the mathematical software named Maple, so most Answers will have some Maple code. The function fsolve is a command of the Maple language that applies the numeric techniques discussed in the last paragraph. There is similar functionality in Excel.

@Remalisa Do you expect Xa, Xb, Ya, Yb (and we may as well include Xc and Yc) to be scalar variables, i.e., to have specific numeric values, rather than to be functions (of t)?

Let's describe the set of information that you have: 5 second-order ODEs with the same functional variable varphi(t), 6 presumably scalar variables, and only 1 initial condition varphi(0) = 0. Is that right?

@pooyan1990 How long does it run, roughly, before you get the error message? Less than a second? Several seconds? Several minutes?

Your initial condition is the most complicated that I've ever seen on a differential equation, and it's amazing that dsolve can solve it even when you give numeric values to those three symbols, because C and especially theta are still symbolic.

An assuming clause applies the assumptions only to the command to which it's attached. If instead you use an assume command, the assumptions will always be in effect. This may allow you to get a fully symbolic solution if you have enough time and memory. First free up as much memory as possible on your system. Make your first command after restart

assume(B/L__1 >= 2, B > 0, H > 0, L__1 > 0);

and keep everything else the same. I let the dsolve run until it consumed 6 Gig memory. If I'd let it run longer, maybe I would've gotten a solution.
 

@vv You are computing the the integral from -1 to 1, wheras the problem says 1 to -1. And how do you know that you're going along the bottom half? The integral is not independent of path: The integral clockwise around the unit circle is 4*I, not 0.

@mmcdara What you describe is the default behavior of Heaviside at 0; that default can be changed. According to ?Heaviside

• When converting Heaviside to piecewise you may want to have the conversion performed taking Heaviside(0) = 1. For that purpose set the Environment Variable _EnvUseHeavisideAsUnitStep to true....

That variable declarations (such as local) can be given initial values is NOT an undocumented feature: It's documented in section 6.5 of the Programming Guide, which is available in the built-in help system. Indeed, you can also give an initial value to an export or a global or give a default value to a proc parameter using nearly identical syntax. 

Do you have some reason to believe that such a k exists?

I suspect that when your instructor gave you this assignment, there was a diagram showing what was meant by "the confined region." Is it something like this?

Hint: Each of the procedures list_to_set, set_to_list, PowerSet, and RandSet can be composed as a single arrow expression using only two neatly formatted lines, one for header and one for body, with no line exceeding half the width of the screen. I give this hint to express the simplicity of the exercises, not to encourage you to write it as short as possible. So, if it takes you 5 lines, fine; but if you find yourself not done at 20 lines, you're probably going about it wrong and should post your partial code and ask for help.

One implication of the above is that no local variables, no global variables, and no assignment statements are needed; however, if you want to use local variables and assignment statements, there's nothing wrong with that.

I'm sorry, but as a Moderator here I felt that I couldn't let such a shoddy^[*1] piece of work remain as a Post, so I made it a Question. As such, some of its inaccuracies and misconceptions can be addressed with proper Answers.

[*1] It's shoddy in terms of English usage, Maple usage, layout, and mathematics.

@mehdi jafari Such a function is 1 - Heaviside(x).

@vv Thank you, VV. Part of the problem is that types and properties get conflated when they have the same name, as with rational, and is falls back to checking the type only. Certainly type(Pi + exp(1), rational) is correctly false.

Compare your sqrt(2) examples with 

simplify((cos(sqrt(2)) + I*sin(sqrt(2)))^infinity);
                                    undefined + undefined*I

@acer Thank you, Acer; that's a wealth of specific technical information.

I think that you'll need to define "min max algebra" or provide a link to a reference before you'll get an Answer.