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separable...

vv 13837
July 23 2017
0 0

@nm 

The new ode is solved via the Bernoulli method (use infolevel). Forcing "separable" ==>

dsolve(diff(y(x), x) = 3*x^2*y(x)^2 , y(x), ['separable']);

    

 

Standard...

vv 13837
July 22 2017
0 0

@Adam Ledger 

Sorry but the term is a standard one. See https://en.wikipedia.org/wiki/Pathological_(mathematics)

pathological function...

vv 13837
July 22 2017
0 0

@Adam Ledger 

f(x) = x^2*ln(x),  f(0) = 0

is a "pathological" function; it is of class C^1 but not C^2  (f'' is unbounded at 0 and actually f''(0+) = - infinity).

Note that for such functions, the significance of O(...)  in Maple is different from the Landau symbol
(in the sense that sub-polynomial terms may be included in O, see ?series).
Note also that the new  MultiSeries:-multiseries  uses the standard (Landau) significance for O.

  

Here are the results in Maple 2017.

 

> 

S1:=series(x^2*ln(x), x, 1);

series(+O(x^2),x,2)

 (1)
> 

S2:=series(x^2*ln(x), x, 2);

series(+O(x^2),x,2)

 (2)
> 

S:=series(x^2*ln(x), x, 3);

series(ln(x)*x^2,x)

 (3)
> 

whattype(S); op(0,S);

series

  

x

 (4)
> 

SS:=MultiSeries:-series(x^2*ln(x), x, 5);

series(-ln(1/x)*x^2,x)

 (5)
> 

whattype(SS);  op(0,SS);

series

  

x

 (6)
> 

M:=MultiSeries:-multiseries(x^2*ln(x), x, 5);

SERIES(Scale, [-1/_var[1/ln(1/x)]], 0, algebraic, [2], infinity, integer, _var[x], -_var[x]^2/_var[1/ln(1/x)])

 (7)
> 

whattype(M); op(0,M);

function

  

SERIES

 (8)
> 

diff(x^2*ln(x), x);

2*x*ln(x)+x

 (9)
> 

diff(x^2*ln(x), x,x);

2*ln(x)+3

 (10)
> 

 

equivalence...

vv 13837
July 18 2017
0 1

@rlopez 

It depends of course on how the residuals are computed in the inplicit approach.
But the space is finite dimensional and so all the norms are equivalent (for linear models).
The main problem is here the model which I think is not adequate; even y = ax+b is much better.

model...

vv 13837
July 18 2017
0 0

@rlopez

No need for implicit functions here because the fitting curve can be inversed x = (y - A*y^3)/B, or equivalently  x = a*y - b*y^3.

The problem is that this curve passes thru (0,0) which is far from the provided data, so the "model" seems to be unsuitable.

Student or not student?...

vv 13837
July 17 2017
0 0

@rlopez 

 

I had the impression that Analytic is more robust than a command in the Student package.

It's a disappointment not being so; in the next example Analytic looses roots!

 

> 

h:=sin(x) - 1/(x+1);

sin(x)-1/(x+1)

 (1)
> 

plot(h, x=0..50);

 
> 

Student:-Calculus1:-Roots(h,x=0..50, numeric);

[.6507516780, 2.880986321, 6.418396937, 9.327799981, 12.63975157, 15.64785930, 18.89982878, 21.94755715, 25.17096082, 28.24012763, 31.44675114, 34.52936974, 37.72493787, 40.81678838, 44.00451897, 47.10309963]

 (2)
> 

nops(%);

16

 (3)
> 

RootFinding:-Analytic(h, x, 0-I/10 .. 50+I/10):
> 

sort([%]);

[.650751677964220, 2.88098632105449, 12.6397515714604, 21.9475571476892, 25.1709608227370, 28.2401276349762, 31.4467511436534, 34.5293697434195, 37.7249378670212, 40.8167883764839, 44.0045189699563, 47.1030996250431]

 (4)
> 

nops(%)

12

 (5)
> 

 

> 

# Strangely, increasing the imaginary part ==> OK.
##################################################
> 

RootFinding:-Analytic(h, x, 0-I .. 50+I):
> 

sort([%]);

[.650751677964220, 2.88098632105449, 6.41839693725700, 9.32779998112870, 12.6397515714604, 15.6478593040306, 18.8998287837776, 21.9475571476892, 25.1709608227370, 28.2401276349762, 31.4467511436534, 34.5293697434195, 37.7249378670212, 40.8167883764839, 44.0045189699563, 47.1030996250431]

 (6)
> 

nops(%)

16

 (7)
> 

 

 

 
 

 

Not a solution...

vv 13837
July 15 2017
0 0

@Rouben Rostamian  

But actually Maple is wrong:  sol[1] (the red one) is not a solution! ( dy/dx should be >=0).

Edit. The solution of the Cauchy problem y(0)=a is unique (x >= 0)  for a > 0.
For a=0  there are two solutions: y = 0 and y = x^3 / 9 (Maple finds only y=0).

facm...

vv 13837
July 14 2017
0 0

@quo 

factrix ignores the second argument. The factor is computed by the procedure.

If you already know the factor, use this simple one:

 

> 

facm:=proc(A::Matrix, q::algebraic)
  local AA,qq;
  subs([qq=q,AA=1/q*A], qq*AA)
end proc;

proc (A::Matrix, q::algebraic) local AA, qq; subs([qq = q, AA = A/q], qq*AA) end proc

 (1)
> 

M:=Matrix([[0,-2*m*omega,0], [2*m*omega,0,0], [0,0,0]]);

_rtable[18446744074366181238]

 (2)
> 

facm(M,2);

2*_rtable[18446744074366175342]

 (3)
> 

facm(M,m);

m*_rtable[18446744074366169206]

 (4)

slope?...

vv 13837
July 13 2017
0 1

But whose slope?

But ......

vv 13837
July 11 2017

But in your approximations appears gamma itself, for which evalf is used!

I think that "solid" people pa...

vv 13837
July 07 2017

I think that "solid" people parametrize and obtain the result if they want. Thank you for your thanks.

@Markiyan Hirnyk 

other tools...

vv 13837
July 07 2017

@Markiyan Hirnyk 

You need other tools for such problems. E.g. for the area:

f:=(1/10)*x^2-(69/790)*x+5569/255960-(3/79)*x*y-(77/4266)*y+(1/4)*y^4-(1/3)*y^3+(1/6)*y^2 - 3/1330:
evalf(Int( piecewise(f<0,1,0), x=0 .. 1, y=0..1));

       .1706012052

For a symbolic result, parametrize first. ==>

 

 

 

 

 

convert does not replace a space by \n; ...

vv 13837
July 07 2017
0 0

convert does not replace a space by \n; it simply inserts some \n 's.

Here is a test:


 

> 

restart;
> 

a:=add(f(i),i=1..1000000):
> 

s:=convert(a,string):
> 

length(s);

9888904

 (1)
> 

fprintf("d:/dnld/ss.txt","%s\n",s);

9888905

 (2)
> 

s1:=StringTools:-DeleteSpace(s):
> 

length(s1);

9888895

 (3)
> 

fprintf("d:/dnld/ss1.txt","%s\n",s1);

9888896

 (4)
> 

close("d:/dnld/ss.txt");
> 

close("d:/dnld/ss1.txt");
> 

#####################
> 

restart;
> 

s:=readbytes("d:/dnld/ss.txt",infinity,TEXT):
> 

length(s);

9888905

 (5)
> 

f:=x->x;

proc (x) options operator, arrow; x end proc

 (6)
> 

eval(parse(s));

500000500000

 (7)
> 

s1:=readbytes("d:/dnld/ss1.txt",infinity,TEXT):
> 

eval(parse(s1));

500000500000

 (8)
> 

 

 

 

Just for fun...

vv 13837
July 06 2017

Just for fun, let's find for how many lists is the result of addList correct.
Denote by f(n) the number of "correct" lists with n elements.
Obviously such a list must have the entries in {±1, ±2,...,, ±n}.

f:=proc(n::posint)
   local v, i, P:=Iterator:-CartesianProduct([seq(1..n),seq(-n..-1)]$n);
   add(`if`(add(v) = add(v[v[i]],i=1..n),1,0), v=P) 
   end:

seq('f'(k)=f(k),k=1..6);
     f(1) = 2, f(2) = 12, f(3) = 78, f(4) = 776, f(5) = 9520, f(6) = 172932

It seems that the sequence f(n) is a serious candidate for  https://oeis.org/

 

 

t=x was just a guess. y must be >0 if...

vv 13837
July 04 2017
0 0

t=x was just a guess.
y must be >0 if the solution is supposed to pe real (otherwise y^(25/6) is complex in general).
You will need numeric solutions, so the value for x and the initial condition must be given.
[I suspect that the ODE does not have real solutions computable by Maple].

@mehdi jafari 

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