## 5093 Reputation

16 years, 210 days
Munich, Germany

## sign...

I think that Maple's sign is correct, for k=1 the numerical answer is  - 1.5575878673834*I (one can use MMA online, which does not know the symbolic answer, but finds a numerical result)

## Example ?...

What do you mean by that - can you give a specific example for that?

## solutions...

a=0.573179148515927e-1, v = 27.0772299601075 or
a=9.54497166461128*10^14, v = .811477602357159

seem to be the only solutions

## Speedtest...

I have Maple 2017.3 32 bit and get similar results (5 times faster, 10 times for accuracy)

## Alternative: Shaw & Brickman...

Neat :-)

I once used William T. Shaw, Nick Brickman, http://arxiv.org/abs/0901.0638

## "recover"...

Yes, simplifying diff(solution)/integrand gives 1.

Note that d=1 (see above) and a=1 (using cNew= c/a instead) can be assumed [and perhaps some of the assumptions can be weakend]

## usage...

I think it is more clear than it displays with all the parameters:

The 1st means: p = constant1 * q where this constant1 depends on (... what you easily see ...).

The 2nd should be read as constant2 * q = 0 (one has to know that Maple abbreviates a bit here), where this constant2 depends on your parameters).

So you have to discuss for which parameters that constant2 equals 0, if you want solutions beyond q=0=p.

## example ?...

You may post / upload one or two examples for which you have problems.

## why ?...

This equation can be solved 'directly', exact.

## approximation close to 0...

For small values (say below 1E-4) the value is approximated by
1.23107661228290 - 0.553095709523571*log(Vzero)

## denominators...

I always try to get rid of denominators (just multiply [and check finally]), here giving the linear system - for which fsolve works without pain.

## yes...

Yes, for example used by Bourbaki (btw deg <= ... ) for the degree

## double precision...

Guessing you have in mind some compiled version (k=352000000, Digits=18 for presentations of IEEE doubles <-> decimals ):

2*k*Pi 'is' 1159557927732361/2251799813685248 * pow(2,32) and sinus of it (with higher precision) is -0.811500562932876925e-8 or -2452609957903403/4503599627370496 * pow(2,-26)

avoiding any "equation labels"

## file names (2)...

Testing for "%add.mw" (Win 7, Maple 32 bit): I do not have Maple 2018, but for Maple 2017 it fails  and for Maple 18 and Maple 2016 it works. For classical sheets *.wms it works in all cases. A crosscheck with the "DOS prompt" shows no unexpected characters (as it sometimes happens for files from an Apple machine)

## With some modification "split and s...

With some modification "split and symmetry" works to show it is zero:

# using symmetry in Pi/2
2*Int(cos(2*x)/(1+2*sin(3*x)^2), x = 0 .. Pi/2);
Change(%, 2*x=xx,xx): subs(xx=x, %): combine(%);

# split and reduce to common range
Split(%, [Pi/3, 2*Pi/3]);
op(1, %) + Change(op(2, %), x = Pi/3 + w) + Change(op(3, %), x = 2*Pi/3 + w);
combine(%): expand(%): combine(%);

# which is zero by symmetry of the integrand
op(1, %):
'eval(%, w = Pi/6 - w) = -eval(%, w = Pi/6 + w)';
expand(%): is(%);

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