@vv You're likely right about the OP's desired output, and this is not meant as a criticism of your Reply, but the commands Normal and Rem aren't doing anything special here with respect to finite-field arithmetic. The results are the same as would be obtained by doing the symbolic algebra over the rationals and then applying mod 13 to the coefficients alone. To wit:

(x-1)/(x-3) mod 13;
             (x + 12)/(x + 10)

r:= rem(x-1, x-3, x, 'q'):  q + r/(x-3) mod 13;
              1 + 2/(x + 10)

Note to the OP: Both VV and I have used q here for "quotient". This is not meant to be confused with q the finite-field element used elsewhere in this thread.

Thanks for this app: It's great to have a calculator form of Maple on my phone.

Is there a document or help page that tells what mathematical functions are implemented? While Answering a Question here last night, I discovered that erf is implemented if entered from the alphabetic keyboard, but BesselJ is not.

I hope that you'll add a feature to "lock on" the alternate keyboards. Otherwise, I need to select the alphabetic keyboard for each letter of B-e-s-s-e-l-J.

Also, it'd be great if it remembered previously entered expressions (simply in a most-recents list, as other Android calculators do).

@HS Okay, Expand is not the right command for this; it expects a polynomial argument. Let's suppose that there was a command that did what you wanted. I'll call it ModOut. What would be your expected output of

ModOut((x-1)/(x-3)) mod 13;

?

Do you have a factorization algorithm that you want to implement in Maple? Or were you planning to use Maple's ifactor?

@ I solved it both in WolframAlpha and Maple, then compared the results numerically using Maple, and I decided they were equal.

Do you have Maple? If so, you can look up the help on StruveH and LommelS1. I don't have time at the moment, but it may be possible to convert these to a more-familiar form using Maple's FunctionAdvisor and related tools. It might help if x can be restricted (to, say, x > 0, for example).

@Aisyah In situations where there is an extra equation, we would hope that there is a redundant equation---one that can be derived from some subset of the others. If we find and remove that equation, it doesn't change the system.

Unfortunately, that's not the case here: Eq2 and Eq3 are directly contradictory; at most one of them can be correct. To see this, do diff(Eq3, eta) and compare the result with Eq2. You'll see that 3, but not all 4, of the signs are different. So, you'll need to carefully review your source material to see where the mistake was made. It's probably just a change-of-sign error.

If you decide to remove Eq2, there's no longer any 2nd derivative of G in the system, so you need to remove the boundary condition on D(G) (which is the first of bcs2).

Then solve and plot solutions only for phi = 0. There's some more subtlety involved in solving for the other values of phi (that "Newton iteration is not converging" stuff), which we'll look at after having the phi = 0 solution. I think we may need to use that solution as a "bootstrap" to get the others.

@tomleslie You're right: I grossly underestimated the true size of the OP's plot. The options add a lot that I didn't account for. In addition to CodeTools:-Usage, it would be useful comparative information to apply length(...) to each plot.

Doing that:

• my 300x300 plot3d: 90,230 (roughly 300²)
• the OP's 500x70 matrixplot: 18,828,337
• your 500x70 matrixplot: 25,055,345 (!!!)
• exact same matrix in matrixplot with no options at all: 35,088 (roughly 500*70).

So, the OP's matrixplot is about 536 (!) times larger (datawise) than the exact same matrix plotted in matrixplot with no options.

You do have a valid Question about dsolve here, but it has nothing to do with the rkf45 algorithm. This is a boundary-value problem (BVP). The rkf45 algorithm is for initial-value problems (IVPs). At this point, however, you don't need to worry about this; Maple can already figure out that this is a BVP.

@Carl Love WolframAlpha (which is available free online for calculator-type queries, such as your integral) gives the result as 

(-(x*HypergeometricPFQ[{1/2, 1/2}, {3/2, 3/2, 2}, -x^2/4]) + x*HypergeometricPFQ[{1/2}, {3/2, 2}, -x^2/4]*Log[x])/2

That translated into Maple is

x/2*(ln(x)*hypergeom([1/2], [3/2,2], -x^2/4) - hypergeom([1/2,1/2], [3/2,3/2,2], -x^2/4))

Maple 2019 gives the result as (I also applied simplify(..., size))

1/2*Pi*x*(BesselJ(0,x)*StruveH(1,x)-StruveH(0,x)*BesselJ(1,x))*ln(1/2*x)-x*
BesselJ(0,x)*D[1](LommelS1)(1,1,x)+x*BesselJ(1,x)*D[1](LommelS1)(0,0,x)+1/2*(-
Pi*x*(ln(2)-ln(x)-1)*StruveH(0,x)-2*ln(x)-2)*BesselJ(1,x)+1/2*(Pi*(ln(2)-ln(x))
*StruveH(1,x)+2*ln(x))*BesselJ(0,x)*x

A quick numerical check over real ranges confirmed (although it's not iron-clad proof) that these two expressions are equal.
 

Would you please define "URI"?

@HS That depends on whether x is the special field element (defined with alias and RootOf) or just a variable.

Your syntax problem is extra multiplication signs, plainly visible in the plaintext code in your Question. These probably come from extra spaces in 2D Input. In 2D Input a function name (such as Expand) must be immediately (without intervening spaces) to the left of the left parenthesis that begins its arguments. In 1D input, it doesn't matter. 

And do you want to do it in Maple Companion? 

Someone with the power to do so needs to add Maple Maple Companion to the Products list for Questions here.

@Rouben Rostamian  Yes, NextZero can be somewhat unreliable in that regard for functions that have an extreme number of vertical asymptotes, as this one does.

However, my Answer did show 245.3396184 and 481.5561050; you just need to scroll to right. Regarding 530.0902992 and 578.7535346: I'll admit that it looks like the function has roots near those values, but putting those numbers into the function at Digits=15 returns values far from 0.

@Joe Riel Neat trick. I'd only ever considered that for alternate spellings. It should be noted that the way that the unevaluation quotes work in _options['view'] is very special. Using 'view' or 'zoom' elsewhere in the procedure, the quotes have no effect, just like any procedure parameter. However, ':-view' and ':-zoom' act normally.