Would you please attach a picture or PDF of a schematic diagram of the circuit? You could put it in the worksheet, but that's not essential at this point. A hand drawing would be sufficient. 

If I understand your worksheet correctly, then all resistors have 1 unit of resistance, all capacitors have one 1 unit of capacitance, and all inductors have 1 unit of inductance. Is that correct?

@Christopher2222 Add the option quality= 100 to the Write command. This indicates 100% of the original quality. The default is 75.

@nm If we can presume to know that the singularity (i.e., vertical-slope point) is at x= - exp(-1), then we can use the sample option to make sure that x-value is specifically used. Adding the adaptive option makes it plot extra points as the magnitude of the slope increases.

plots:-display(
    plot~(
        [LambertW(x), LambertW(-1, x)], x= -1..4, sample= [-1, -exp(-1), 4],
        color=~ [red, blue], adaptive= 8, view= [-1..4, -3.5..1.5]
    )
);

@mmcdara Was I wrong in saying that the system had derivatives with respect to bn?

It's so curious that there are a large number of both good and bad edges. I think that's a key piece of evidence. Let's look at the numeric plot structures of both types edges and try to figure out the key difference. 

@C_R When we're talking/writing casually of Maple programming, the words "function" and "procedure" are often used interchangeably. But when these words are used as keywords for types in Maple code, they are very different; indeed, they are disjoint. The f in your procedure header is declared function. This won't work. You can make that procedure or appliable (which is a supertype of procedure).

@Christopher2222 Reversing the order of the exponentiations (and, here, just using symbolic variables to emphasize the generality) we have

(x^r)^n = x^(r*n); simplify(%) assuming n::integer;

for any x and r and any integer n.

Your title "Cube root to the third...." states this reverse-order case. I think that you meant the title to be "Cube root of the third power doesn't simplify."

@NurinFYP No, it's not the same error message! The latest error is because you used beta[1] in the equations when you should've used beta__1.

All of your errors have been very simple errors that I could spot in seconds without even seeing your actual worksheet. I'm not saying that you should be able to spot them in seconds, but these trivial typos are not something worth giving up about.

@Jamie128 Surprisingly, I don't think that there's a simple way to get forget to forget everything (i.e., all "remember tables").

@NurinFYP You should change all the uppercase Zeta in your equations to lowercase zeta, or simply use another variable. 

@NurinFYP I think "global variables" is a bit of a misnomer. What it means here are "unassigned parameters"---variables representing fixed numbers whose numeric values haven't been assigned yet. They are allowed if you declare them with a parameters option. But in this case, I think it's just because of a typo. You assigned the value of ic4 where I think you meant ic3.

There is nothing special about forget that makes it unsuitable for use in a loop. Is there something from its help page that you're having trouble understanding?

You said that there are 3 nonzero entries per row. Is the matrix tridiagonal (i.e., A[i,j] <> 0 implies |i - j| <= 1)? Are there symbolic variables in the entries? If so, how many distinct variables are there? Do they occur only in rational functions (quotients of polynomials)? Are the numbers all rational numbers?

The Fortran program shown is stated to be a translation of an Algol program. Try to find that Algol program. Algol is much closer to Maple than C is.

To use FindTransitiveOrientation, set kernelopts(opaquemodules= false), then call GraphTheory:-FindTransitiveOrientation. You must use the module prefix GraphTheory:-, not the with command.

However, why not just call 

GraphTheory:-IsComparabilityGraph(..., transitiveorientation)