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Rouben Rostamian

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These are answers submitted by Rouben Rostamian

Why do you expect a^(3/2) to be the only...

Rouben Rostamian 7711
March 20 2024
2 0

Why do you expect a^(2/3) to be the only solution?  Suppose a = -1.  Then we have

> 

restart;
> 

eq := x^(3/2) = -1;

x^(3/2) = -1

> 

sol := solve(eq);

(1/4)*(I*3^(1/2)-1)^2, (1/4)*(1+I*3^(1/2))^2

> 

simplify(subs(x=sol[1], eq));

-1 = -1

> 

simplify(subs(x=sol[2], eq));

-1 = -1

Add a pointplot...

Rouben Rostamian 7711
March 09 2024
3 0

If you are specifying the x-range as x=a..b,  then you want to include the point with coordinates (a,0) in your graph.  So something like this should work:

with(plots):
display(
   plot(10 + sin(x), x=3*Pi..4*Pi),
   pointplot([3*Pi,0], symbol=point)
);

 

PDF graphics...

Rouben Rostamian 7711
March 03 2024
1 4

Maple's native export to PDF is useless since it embeds the exported drawing within a letter-size (i.e., 8.5in by 11in) document.

Maple's export as EPS (Encapsulated PostScript) works fine with line drawings such as that you have shown.  Save your graphics as EPS and then apply a conversion utility, outside of Maple, to the saved file to produce the PDF equivalent.

There are many such conversions utilities available for free.  The choice depends on your operating system.  I use epstopdf on Linux. 

PS: I might have misunderstood your question.  If by "3D PDF" you mean a PDF document meant to be viewed with the help of 3D glasses, then my answer is totally irrelevant.  Ignore it.

Perhaps a typo?...

Rouben Rostamian 7711
February 21 2024
2 2

Your function F is negative near phi=0 as we see in the graph below.  Then 1/sqrt(F) is complex and therefore your x is complex.  Is that what you expect?  If not, be sure that you have entered F correctly.

> 

restart;
> 

F:= phi -> 3.924999 - 0.024999/sqrt(1 - 2*phi) - 3.900/(1 - phi/6)^(3/2) - 1.648094618*10^(-14)*sqrt(3)*sqrt(1836)*(((1.3972 + sqrt(3)*sqrt(1836))^2 + 3672*phi)^(3/2) - (1.3972 + sqrt(3)*sqrt(1836))^3 - ((1.3972 - sqrt(3)*sqrt(1836))^2 + 3672*phi)^(3/2) + ((1.3972 - sqrt(3)*sqrt(1836))^2)^(3/2))-(1/18)*(sqrt(3)*sqrt(300)*(((1.4472 + sqrt(3)*sqrt(300)/300)^2 - 2*phi)^(3/2) - (1.4472 + sqrt(3)*sqrt(300)/300)^3 - ((1.4472 - sqrt(3)*sqrt(300)/300)^2 - 2*phi)^(3/2) + (1.4472 - sqrt(3)*sqrt(300)/300)^3));

proc (phi) options operator, arrow; 3.924999-0.24999e-1/sqrt(1-2*phi)-3.900/(1-(1/6)*phi)^(3/2)-0.1648094618e-13*sqrt(3)*sqrt(1836)*(((1.3972+sqrt(3)*sqrt(1836))^2+3672*phi)^(3/2)-(1.3972+sqrt(3)*sqrt(1836))^3-((1.3972-sqrt(3)*sqrt(1836))^2+3672*phi)^(3/2)+((1.3972-sqrt(3)*sqrt(1836))^2)^(3/2))-(1/18)*sqrt(3)*sqrt(300)*(((1.4472+(1/300)*sqrt(3)*sqrt(300))^2-2*phi)^(3/2)-(1.4472+(1/300)*sqrt(3)*sqrt(300))^3-((1.4472-(1/300)*sqrt(3)*sqrt(300))^2-2*phi)^(3/2)+(1.4472-(1/300)*sqrt(3)*sqrt(300))^3) end proc

> 

plot(F(phi), phi=-1..1);

 

Download plot-of-F.mw

Integration...

Rouben Rostamian 7711
February 17 2024
2 5

 

> 

restart;

We wish to evaluate

> 

Int(abs(sin(x^4))/(sqrt(x) + x^2), x=0..infinity);

Int(abs(sin(x^4))/(x^(1/2)+x^2), x = 0 .. infinity)

The change of variable x^4 = u converts the integral to

> 

Int(abs(sin(u))/((u^(1/8) + u^(1/2))*(4*u^(3/4))), u=0..infinity);

Int((1/4)*abs(sin(u))/((u^(1/8)+u^(1/2))*u^(3/4)), u = 0 .. infinity)

The integrand is O(1/u^(5/4)) as proc (u) options operator, arrow; infinity end proctherefore the integral converges, albeit slowly.

This is what the integrand looks like:

> 

plot(abs(sin(u))/(4*(u^(1/8) + sqrt(u))*u^(3/4)), u=0..10*Pi);

The abs(sin(u)) factor is Pi-periodic. Accordingly, we split the integration into a sum of

integrations over intervals of length Pi:

> 

J := n -> int(abs(sin(u))/((u^(1/8) + u^(1/2))*(4*u^(3/4))),
        u = n*Pi .. (n+1)*Pi, numeric );

proc (n) options operator, arrow; int((1/4)*abs(sin(u))/((u^(1/8)+u^(1/2))*u^(3/4)), u = n*Pi .. (n+1)*Pi, numeric) end proc

> 

J(0), J(1), J(2), J(3), J(1000);

.2311384906, 0.4737708191e-1, 0.2625581792e-1, 0.1783240068e-1, 0.2025655105e-4

Here is the result:

> 

add(J(n), n=0..1000);

.5348206907

Take more terms for greater accuracy.

 

> 

 

Aside: I attempted to do the change of variables through Maple's IntegrationTools:-Change

but the result was wrong.  Perhaps a bug?

> 

Int(abs(sin(x^4))/(sqrt(x) + x^2), x=0..infinity);
IntegrationTools:-Change(%, x=u^(1/4));

Int(abs(sin(x^4))/(x^(1/2)+x^2), x = 0 .. infinity)

-(1/4)*(Int((-u^(11/4)+u^(7/8)+u^(13/8)+u^(19/8)-u^(1/2)-u^(5/4)-u^2+u^(1/8))*abs(sin(u))/(u*(u^3-1)), u = 0 .. infinity))

> 

 

 

Download integration.mw

 

Try this...

Rouben Rostamian 7711
February 16 2024
0 3

Try this (and also read the help page on plot,options).

plots:-display(
	plot(x^2, x=-1..1,   linestyle=dot),
	plot(x^2+1, x=-1..1, linestyle=spacedash),
	plot(x^2+2, x=-1..1, linestyle=dashdot),
thickness=3);

 

Remove the quotes...

Rouben Rostamian 7711
February 16 2024
0 0

The construction `A=70` (with the back-quotes) builds a name consisting of those four characters, just like AB70 is a name consisting of four characters.  The equal sign has no special meaning in that name.  If instead you do

lst := [A=70, B=17, C=27];

(without the quotes,) then

tt := convert(lst, table);

will do what you want.

Ill-posed problem...

Rouben Rostamian 7711
February 07 2024
3 0

This problem is not well-posed.  See the discussion here.

That may not be a meaningful operation...

Rouben Rostamian 7711
January 27 2024
2 2
 

What is the meaning of delta(x^2+y^2-1)?

The Dirac delta is defined through the property
int(delta(x)*f(x), x = -infinity .. infinity) = f(0)    for all continuous functions f.

 

Let's see if we can make sense of something simpler than your

exmaple, say delta(x^3).

Make a change of variables x^3 = u and calculate
"(∫)[-infinity]^(infinity)delta(x^(3))f(x) ⅆx= (∫)[-infinity]^(infinity)delta(u)f(u^(1/3)) (ⅆu)/(3 x^(2))=(∫)[-infinity]^(infinity)delta(u)[(f(u^(1/3)) )/(3 u^(2/3))] ⅆu".

To continue, we need to evaluate the expression within the square brackets

at u = 0 which may not be feasible for all f and therefore delta(x^3) makes no

sense.

Will this do?...

Rouben Rostamian 7711
January 07 2024
1 2 
restart;
with(plots):
with(geom3d):
ball := draw(TruncatedIcosahedron(``)(color="Green"));

Series calculation...

Rouben Rostamian 7711
November 23 2023
0 0

This worksheet shows how to calculate the first five terms of the desired series.  I have coded it as 5 similar blocks.  They may be combined into single for-loop to compute any number of terms.

The coefficient of u^3 in what you have shown seems to be in error.  The (a-1)(a+3) should be (a-1)(a+2)..

 

We have
z = int(sinh(a*sin(x*y)), y = 0 .. 1)

This defines z as a function of x but we want x as a function of z,

as in x(z) so we calculate the Maclaurin series of x(z).
Observe that x = 0 implies that z = 0, and therefore

x(0) = 0.  That gives us the first term of the series.

We find the remaining terms by repeated differentiations.

> 

restart;
> 

eq[0] := z = int(sinh(a*sin(x(z)*y)), y=0..1);

z = int(sinh(a*sin(x(z)*y)), y = 0 .. 1)

> 

rels[0] := x(0)=0;

x(0) = 0

> 

eq[1] := diff(eq[0], z);
convert(%, D);
eval(%, z=0);
eval(%, {rels[0]});
isolate(%, D(x)(0));
rels[1] := rels[0], simplify(%);

1 = int(a*(diff(x(z), z))*y*cos(x(z)*y)*cosh(a*sin(x(z)*y)), y = 0 .. 1)

1 = int(a*(D(x))(z)*y*cos(x(z)*y)*cosh(a*sin(x(z)*y)), y = 0 .. 1)

1 = int(a*(D(x))(0)*y*cos(x(0)*y)*cosh(a*sin(x(0)*y)), y = 0 .. 1)

1 = (1/2)*a*(D(x))(0)

(D(x))(0) = 2/a

x(0) = 0, (D(x))(0) = 2/a

> 

eq[2] := diff(eq[1], z):
convert(%, D);
eval(%, z=0);
eval(%, {rels[1]});
isolate(%, (D@@2)(x)(0));
rels[2] := rels[1], simplify(%);

0 = int(a*((D@@2)(x))(z)*y*cos(x(z)*y)*cosh(a*sin(x(z)*y))-a*(D(x))(z)^2*y^2*sin(x(z)*y)*cosh(a*sin(x(z)*y))+a^2*(D(x))(z)^2*y^2*cos(x(z)*y)^2*sinh(a*sin(x(z)*y)), y = 0 .. 1)

0 = int(a*((D@@2)(x))(0)*y*cos(x(0)*y)*cosh(a*sin(x(0)*y))-a*(D(x))(0)^2*y^2*sin(x(0)*y)*cosh(a*sin(x(0)*y))+a^2*(D(x))(0)^2*y^2*cos(x(0)*y)^2*sinh(a*sin(x(0)*y)), y = 0 .. 1)

0 = (1/2)*a*((D@@2)(x))(0)

((D@@2)(x))(0) = 0

x(0) = 0, (D(x))(0) = 2/a, ((D@@2)(x))(0) = 0

> 

eq[3] := diff(eq[2], z):
convert(%, D);
eval(%, z=0);
eval(%, {rels[2]});
isolate(%, (D@@3)(x)(0));
rels[3] := rels[2], simplify(%);

0 = int(a*((D@@3)(x))(z)*y*cos(x(z)*y)*cosh(a*sin(x(z)*y))-3*a*((D@@2)(x))(z)*y^2*(D(x))(z)*sin(x(z)*y)*cosh(a*sin(x(z)*y))+3*a^2*((D@@2)(x))(z)*y^2*cos(x(z)*y)^2*(D(x))(z)*sinh(a*sin(x(z)*y))-a*(D(x))(z)^3*y^3*cos(x(z)*y)*cosh(a*sin(x(z)*y))-3*a^2*(D(x))(z)^3*y^3*sin(x(z)*y)*cos(x(z)*y)*sinh(a*sin(x(z)*y))+a^3*(D(x))(z)^3*y^3*cos(x(z)*y)^3*cosh(a*sin(x(z)*y)), y = 0 .. 1)

0 = int(a*((D@@3)(x))(0)*y*cos(x(0)*y)*cosh(a*sin(x(0)*y))-3*a*((D@@2)(x))(0)*y^2*(D(x))(0)*sin(x(0)*y)*cosh(a*sin(x(0)*y))+3*a^2*((D@@2)(x))(0)*y^2*cos(x(0)*y)^2*(D(x))(0)*sinh(a*sin(x(0)*y))-a*(D(x))(0)^3*y^3*cos(x(0)*y)*cosh(a*sin(x(0)*y))-3*a^2*(D(x))(0)^3*y^3*sin(x(0)*y)*cos(x(0)*y)*sinh(a*sin(x(0)*y))+a^3*(D(x))(0)^3*y^3*cos(x(0)*y)^3*cosh(a*sin(x(0)*y)), y = 0 .. 1)

0 = -2/a^2+2+(1/2)*a*((D@@3)(x))(0)

((D@@3)(x))(0) = -2*(-2/a^2+2)/a

x(0) = 0, (D(x))(0) = 2/a, ((D@@2)(x))(0) = 0, ((D@@3)(x))(0) = (-4*a^2+4)/a^3

> 

eq[4] := diff(eq[3], z):
convert(%, D);
eval(%, z=0);
eval(%, {rels[3]});
isolate(%, (D@@4)(x)(0));
rels[4] := rels[3], simplify(%);

0 = int(a*((D@@4)(x))(z)*y*cos(x(z)*y)*cosh(a*sin(x(z)*y))-4*a*((D@@3)(x))(z)*y^2*(D(x))(z)*sin(x(z)*y)*cosh(a*sin(x(z)*y))+4*a^2*((D@@3)(x))(z)*y^2*cos(x(z)*y)^2*(D(x))(z)*sinh(a*sin(x(z)*y))-3*a*((D@@2)(x))(z)^2*y^2*sin(x(z)*y)*cosh(a*sin(x(z)*y))-6*a*((D@@2)(x))(z)*y^3*(D(x))(z)^2*cos(x(z)*y)*cosh(a*sin(x(z)*y))-18*a^2*((D@@2)(x))(z)*y^3*(D(x))(z)^2*sin(x(z)*y)*cos(x(z)*y)*sinh(a*sin(x(z)*y))+3*a^2*((D@@2)(x))(z)^2*y^2*cos(x(z)*y)^2*sinh(a*sin(x(z)*y))+6*a^3*((D@@2)(x))(z)*y^3*cos(x(z)*y)^3*(D(x))(z)^2*cosh(a*sin(x(z)*y))+a*(D(x))(z)^4*y^4*sin(x(z)*y)*cosh(a*sin(x(z)*y))-4*a^2*(D(x))(z)^4*y^4*cos(x(z)*y)^2*sinh(a*sin(x(z)*y))+3*a^2*(D(x))(z)^4*y^4*sin(x(z)*y)^2*sinh(a*sin(x(z)*y))-6*a^3*(D(x))(z)^4*y^4*sin(x(z)*y)*cos(x(z)*y)^2*cosh(a*sin(x(z)*y))+a^4*(D(x))(z)^4*y^4*cos(x(z)*y)^4*sinh(a*sin(x(z)*y)), y = 0 .. 1)

0 = int(a*((D@@4)(x))(0)*y*cos(x(0)*y)*cosh(a*sin(x(0)*y))-4*a*((D@@3)(x))(0)*y^2*(D(x))(0)*sin(x(0)*y)*cosh(a*sin(x(0)*y))+4*a^2*((D@@3)(x))(0)*y^2*cos(x(0)*y)^2*(D(x))(0)*sinh(a*sin(x(0)*y))-3*a*((D@@2)(x))(0)^2*y^2*sin(x(0)*y)*cosh(a*sin(x(0)*y))-6*a*((D@@2)(x))(0)*y^3*(D(x))(0)^2*cos(x(0)*y)*cosh(a*sin(x(0)*y))-18*a^2*((D@@2)(x))(0)*y^3*(D(x))(0)^2*sin(x(0)*y)*cos(x(0)*y)*sinh(a*sin(x(0)*y))+3*a^2*((D@@2)(x))(0)^2*y^2*cos(x(0)*y)^2*sinh(a*sin(x(0)*y))+6*a^3*((D@@2)(x))(0)*y^3*cos(x(0)*y)^3*(D(x))(0)^2*cosh(a*sin(x(0)*y))+a*(D(x))(0)^4*y^4*sin(x(0)*y)*cosh(a*sin(x(0)*y))-4*a^2*(D(x))(0)^4*y^4*cos(x(0)*y)^2*sinh(a*sin(x(0)*y))+3*a^2*(D(x))(0)^4*y^4*sin(x(0)*y)^2*sinh(a*sin(x(0)*y))-6*a^3*(D(x))(0)^4*y^4*sin(x(0)*y)*cos(x(0)*y)^2*cosh(a*sin(x(0)*y))+a^4*(D(x))(0)^4*y^4*cos(x(0)*y)^4*sinh(a*sin(x(0)*y)), y = 0 .. 1)

0 = (1/2)*a*((D@@4)(x))(0)

((D@@4)(x))(0) = 0

x(0) = 0, (D(x))(0) = 2/a, ((D@@2)(x))(0) = 0, ((D@@3)(x))(0) = (-4*a^2+4)/a^3, ((D@@4)(x))(0) = 0

> 

eq[5] := diff(eq[4], z):
convert(%, D);
eval(%, z=0);
eval(%, {rels[4]});
isolate(%, (D@@5)(x)(0));
rels[5] := rels[4], simplify(%);

0 = int(10*a^3*((D@@3)(x))(z)*y^3*cos(x(z)*y)^3*(D(x))(z)^2*cosh(a*sin(x(z)*y))-40*a^2*((D@@2)(x))(z)*y^4*(D(x))(z)^3*cos(x(z)*y)^2*sinh(a*sin(x(z)*y))+10*a^4*((D@@2)(x))(z)*y^4*cos(x(z)*y)^4*(D(x))(z)^3*sinh(a*sin(x(z)*y))-10*a^4*(D(x))(z)^5*y^5*sin(x(z)*y)*cos(x(z)*y)^3*sinh(a*sin(x(z)*y))-60*a^3*((D@@2)(x))(z)*y^4*(D(x))(z)^3*sin(x(z)*y)*cos(x(z)*y)^2*cosh(a*sin(x(z)*y))+a*((D@@5)(x))(z)*y*cos(x(z)*y)*cosh(a*sin(x(z)*y))-30*a^2*((D@@3)(x))(z)*y^3*(D(x))(z)^2*sin(x(z)*y)*cos(x(z)*y)*sinh(a*sin(x(z)*y))-10*a*((D@@3)(x))(z)*y^3*(D(x))(z)^2*cos(x(z)*y)*cosh(a*sin(x(z)*y))+a^5*(D(x))(z)^5*y^5*cos(x(z)*y)^5*cosh(a*sin(x(z)*y))-15*a*((D@@2)(x))(z)^2*y^3*(D(x))(z)*cos(x(z)*y)*cosh(a*sin(x(z)*y))+a*(D(x))(z)^5*y^5*cos(x(z)*y)*cosh(a*sin(x(z)*y))+5*a^2*((D@@4)(x))(z)*y^2*cos(x(z)*y)^2*(D(x))(z)*sinh(a*sin(x(z)*y))-45*a^2*((D@@2)(x))(z)^2*y^3*sin(x(z)*y)*(D(x))(z)*cos(x(z)*y)*sinh(a*sin(x(z)*y))+15*a^3*((D@@2)(x))(z)^2*y^3*cos(x(z)*y)^3*(D(x))(z)*cosh(a*sin(x(z)*y))+15*a^2*(D(x))(z)^5*y^5*sin(x(z)*y)*cos(x(z)*y)*sinh(a*sin(x(z)*y))-10*a^3*(D(x))(z)^5*y^5*cos(x(z)*y)^3*cosh(a*sin(x(z)*y))+15*a^3*(D(x))(z)^5*y^5*sin(x(z)*y)^2*cos(x(z)*y)*cosh(a*sin(x(z)*y))+10*a*((D@@2)(x))(z)*y^4*(D(x))(z)^3*sin(x(z)*y)*cosh(a*sin(x(z)*y))+30*a^2*((D@@2)(x))(z)*y^4*(D(x))(z)^3*sin(x(z)*y)^2*sinh(a*sin(x(z)*y))-5*a*((D@@4)(x))(z)*y^2*(D(x))(z)*sin(x(z)*y)*cosh(a*sin(x(z)*y))-10*a*((D@@3)(x))(z)*y^2*((D@@2)(x))(z)*sin(x(z)*y)*cosh(a*sin(x(z)*y))+10*a^2*((D@@3)(x))(z)*y^2*cos(x(z)*y)^2*((D@@2)(x))(z)*sinh(a*sin(x(z)*y)), y = 0 .. 1)

0 = int(10*a^3*((D@@3)(x))(0)*y^3*cos(x(0)*y)^3*(D(x))(0)^2*cosh(a*sin(x(0)*y))-40*a^2*((D@@2)(x))(0)*y^4*(D(x))(0)^3*cos(x(0)*y)^2*sinh(a*sin(x(0)*y))+10*a^4*((D@@2)(x))(0)*y^4*cos(x(0)*y)^4*(D(x))(0)^3*sinh(a*sin(x(0)*y))-10*a^4*(D(x))(0)^5*y^5*sin(x(0)*y)*cos(x(0)*y)^3*sinh(a*sin(x(0)*y))-60*a^3*((D@@2)(x))(0)*y^4*(D(x))(0)^3*sin(x(0)*y)*cos(x(0)*y)^2*cosh(a*sin(x(0)*y))+a*((D@@5)(x))(0)*y*cos(x(0)*y)*cosh(a*sin(x(0)*y))-30*a^2*((D@@3)(x))(0)*y^3*(D(x))(0)^2*sin(x(0)*y)*cos(x(0)*y)*sinh(a*sin(x(0)*y))-10*a*((D@@3)(x))(0)*y^3*(D(x))(0)^2*cos(x(0)*y)*cosh(a*sin(x(0)*y))+a^5*(D(x))(0)^5*y^5*cos(x(0)*y)^5*cosh(a*sin(x(0)*y))-15*a*((D@@2)(x))(0)^2*y^3*(D(x))(0)*cos(x(0)*y)*cosh(a*sin(x(0)*y))+a*(D(x))(0)^5*y^5*cos(x(0)*y)*cosh(a*sin(x(0)*y))+5*a^2*((D@@4)(x))(0)*y^2*cos(x(0)*y)^2*(D(x))(0)*sinh(a*sin(x(0)*y))-45*a^2*((D@@2)(x))(0)^2*y^3*sin(x(0)*y)*(D(x))(0)*cos(x(0)*y)*sinh(a*sin(x(0)*y))+15*a^3*((D@@2)(x))(0)^2*y^3*cos(x(0)*y)^3*(D(x))(0)*cosh(a*sin(x(0)*y))+15*a^2*(D(x))(0)^5*y^5*sin(x(0)*y)*cos(x(0)*y)*sinh(a*sin(x(0)*y))-10*a^3*(D(x))(0)^5*y^5*cos(x(0)*y)^3*cosh(a*sin(x(0)*y))+15*a^3*(D(x))(0)^5*y^5*sin(x(0)*y)^2*cos(x(0)*y)*cosh(a*sin(x(0)*y))+10*a*((D@@2)(x))(0)*y^4*(D(x))(0)^3*sin(x(0)*y)*cosh(a*sin(x(0)*y))+30*a^2*((D@@2)(x))(0)*y^4*(D(x))(0)^3*sin(x(0)*y)^2*sinh(a*sin(x(0)*y))-5*a*((D@@4)(x))(0)*y^2*(D(x))(0)*sin(x(0)*y)*cosh(a*sin(x(0)*y))-10*a*((D@@3)(x))(0)*y^2*((D@@2)(x))(0)*sin(x(0)*y)*cosh(a*sin(x(0)*y))+10*a^2*((D@@3)(x))(0)*y^2*cos(x(0)*y)^2*((D@@2)(x))(0)*sinh(a*sin(x(0)*y)), y = 0 .. 1)

0 = 16/3+(16/3)/a^4-(160/3)/a^2+10*(-4*a^2+4)/a^2-10*(-4*a^2+4)/a^4+(1/2)*a*((D@@5)(x))(0)

((D@@5)(x))(0) = -2*(16/3+(16/3)/a^4-(160/3)/a^2+10*(-4*a^2+4)/a^2-10*(-4*a^2+4)/a^4)/a

x(0) = 0, (D(x))(0) = 2/a, ((D@@2)(x))(0) = 0, ((D@@3)(x))(0) = (-4*a^2+4)/a^3, ((D@@4)(x))(0) = 0, ((D@@5)(x))(0) = (16/3)*(13*a^4-10*a^2+13)/a^5

Here is the series:

> 

series(x(z), z);
eval(%, {rels[5]});

series(x(0)+(D(x))(0)*z+((1/2)*((D@@2)(x))(0))*z^2+((1/6)*((D@@3)(x))(0))*z^3+((1/24)*((D@@4)(x))(0))*z^4+((1/120)*((D@@5)(x))(0))*z^5+O(z^6),z,6)

series((2/a)*z+((1/6)*(-4*a^2+4)/a^3)*z^3+((2/45)*(13*a^4-10*a^2+13)/a^5)*z^5+O(z^6),z,6)

> 

 

 

Download series.mw

 

Errors...

Rouben Rostamian 7711
November 11 2023
0 2

You have four PDEs in the five unknowns 

C(X, R, t), R(X, R, t), T(X, R, t), U(X, R, t), V(X, R, t)

Additionally, U(X,R,t) appears just as U in the fourth PDE.  There may be other errors as well.

Suggestions:

  1. Post an actual worksheet rather than copy/paste.  That will help people that are going to help you.
  2. Before you do a composite computation in a for-loop, try just one case in order to detect where the error occurs.  You may add the for-loop afterward once you see that the basic code works.

Is this good enough?...

Rouben Rostamian 7711
November 08 2023
6 9 
restart;
ode1 := diff(x(t),t)=2*x(t)*y(t);
ode2 := diff(y(t),t)=1-x(t)^2-y(t)^2;
DEtools:-DEplot([ode1,ode2],[x(t),y(t)],
	t=0..1,x = -4..4, y = -4..4,
	arrows=curve, linecolor=red, arrowsize=1.5,
	axes=boxed, color='magnitude[legacy]');

Heap size...

Rouben Rostamian 7711
October 19 2023
1 4

The default java heap size is 512.  You may change it through the commandline flag −j, as in

maple -j 65536 -x filename.mw

I don't have an answer to your second question.

Use "Explore"...

Rouben Rostamian 7711
October 15 2023
2 1

Use Explore.  Move the sliders to vary M and k.

> 

restart;
> 

addcoords(z_cylindrical, [z, r, t], [r*cos(t), r*sin(t),z]):
> 

F := -2*M^2*sin(theta)^2 - 2*k*r^2

-2*M^2*sin(theta)^2-2*k*r^2

> 

Explore(plot3d([F,0], r=0..1, theta=0..2*Pi, coords=z_cylindrical), M=-1.0..1.0, k=-1.0..1.0);

 

Download mw.mw

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