## Is there an error in dsolve using method=laplace...

Maple 2021

I was looking at symbolically solving a second-order differential equation and it looks like the method=laplace method has a sign error when the coefficients are presented in a certain way.  Below is a picture of some examples with and without method=laplace that should all have the same closed form.  Note that lines (s6) and (s8) have different signs in the exponential than they should have (which is a HUGE problem):

 >
 >
 (1)
 >
 (2)
 >
 (3)
 >
 (4)
 >
 (5)
 >
 (6)
 >
 (7)
 >
 (8)
 >
 (9)
 >
 (10)
 >
 >
 (11)
 >
 (12)
 >
 (13)
 >
 (14)
 >

## How can the functional derivative/variation of the...

Hello Everyone,

My questions is about a straight forward as my title suggests. I know how to compute the variation by hand and of course it is in plently of books, but I would like to know how I could possibly do it in Maple as I have some terms with non-trivial contributions where I infact need that specific variation.

I have attached my file I have been working with that has more comments.

RicciAction.mw

Thank you

## Cannot solve with dsolve the self similarity probl...

Hello,

I am trying to solve the self similarity problem for external natural convection which is the following coupled problem

•     3/4*F(eta)*Theta(eta)'  = Theta(eta)''
•     1/2 (F(eta)')^2 - 3/4 F(eta)*F(eta)'' = Pr*(Theta(eta) - F(eta)''')

where Pr is a constant and  ()' denotes differentiation with respect to eta. The boundary conditions are:

•     at eta = 0, F = 0, F'=0, Theta = 1
•     at eta -> infinity, F' = 0, Theta = 0

I tried the following but I cannot obtain a solution

 > Pr:=1
 (1)
 > etamax := infinity
 (2)

 > eq:= 3/4*F(eta)*diff(theta(eta),eta) = diff(theta(eta),eta,eta),1/Pr*(1/2*diff(F(eta),eta)^2 - 3/4*F(eta)*diff(F(eta),eta,eta)) = -diff(F(eta),eta,eta,eta) + theta(eta)
 (3)

 >

 > bcs := F(0) = 0, theta(0)=1, D(F)(0)=0,theta(etamax)=0,D(F)(etamax)=0
 (4)

 > sol:=dsolve([eq,bcs],[F(eta),theta(eta)])

## Chevron's lemma...

restart;
with(geometry):
with(plots):
_EnvHorizontalName = 'x':
_EnvVerticalName = 'y':
EQ := proc(M, N)
local eq, sol;
eq := simplify(expand((y - M[2])/(x - M[1]) - (N[2] - M[2])/(N[1] - M[1])));
sol := solve(eq, y);
RETURN(y = sol); end proc:
_local(D);
point(A, [-2, 7]):
point(B, [-5, -2]):
point(C, [8, -7]):
point(E, [1, 4]):
EQ([-5, -2], [8, -7]):
point(D, [1, subs(x = 1, rhs(%))]):
BD := distance(B, D):
DC := distance(C, D):
triangle(ABC, [A, B, C]):
area(ABC):
triangle(ABD, [A, B, D]):
area(ABD):
triangle(EBD, [E, B, D]):
area(EBD):
triangle(EDC, [E, D, C]):
area(EDC):
triangle(AEC, [A, E, C]):
area(AEC):
triangle(ABE, [A, B, E]):
area(ABE):
is(area(ABE)/area(AEC) = BD/DC):
display*([draw*[A(color = black, symbol = solidcircle, symbolsize = 6),
B(color = black, symbol = solidcircle, symbolsize = 6),
C(color = black, symbol = solidcircle, symbolsize = 6),
ABC(color = blue)],
textplot*([[coordinates(A)[], "A"],
[coordinates(B)[], "B"],
[coordinates(C)[], "C3"]],
align = [above, right])],
axes = none,
title = "Lemme du Chevron");
The program simply reproduces display...Why; Thank you.
display*([draw*[A(color = black, symbol = solidcircle, symbolsize = 12), B(color = black, symbol = solidcircle, symbolsize = 12), C(color = black, symbol = solidcircle, symbolsize = 12), ABC(color = blue)], textplot*([[-2, 7, "A"], [-5, -2, "B"], [8, -7, "C3"]], align = [above, right])], axes = none, title = "Lemme du Chevron")

## How can I Simplify in the physics package a tensor...

Hi MaplePrimes,

I am interested in obtaining some gravitational field equations from an action using the FunDiff command. I have been able to write what I think is a pretty short and quick worksheet(with an arbitrary metric) and I am easily able to obtain the Einstein Field Equations. However, when I introduce some new more complicated terms into the action and apply the Simplify command maple does not appear to be able to evaluate and I end up halting the computation. When I specify a metric Maple, for example Schwarzschild, Maple will easily be able to Simplify my expression but it will use metric components during the process. Where what I am interested in is just the Tensor expression with respective indices.

I was wondering if anyone had any thoughts on how I could resolve this.

I have attached the worksheet that I am working with. I look forward to your thoughts/comments.

Thank you.

ActionFieldEquations.mw

## Monge(s circle of an ellipse...

This figure refuses to turn
Fig := proc(t) local a, b, P, Q, N, R, TG, x0, y0, p1, p2, p3, po, tp, sol; a := 11; b := 7; R := sqrt(a^2 + b^2); P := [R*sin(t), R*cos(t)]; x0 := P[1]; y0 := P[2]; TG := (a^2 - x0^2)*(y - y0)^2 + (b^2 - y0^2)*(x - x0)^2 + 2*y0*x0*(x - x0)*(y - y0) = 0; p1 := implicitplot(x^2/a^2 + y^2/b^2 - 1, x = -11 .. 11, y = -7 .. 7, color = blue); p2 := implicitplot(x^2 + y^2 - a^2 - b^2, x = -15 .. 15, y = -15 .. 15, color = blue); p3 := implicitplot(TG, x = -15 .. 15, y = -15 .. 15, color = red); sol := solve({x^2/a^2 + y^2/b^2 - 1 = 0, TG}, {x, y}, explicit); Q := [subs(sol[1], x), subs(sol[1], y)]; N := [subs(sol[2], x), subs(sol[2], y)]; po := plot([P, Q, N], style = point, symbolsize = 15, symbol = solidcircle, color = red); tp := textplot([[P[], "P"], [Q[], "Q"], [N[], "N"]], 'align' = {'above', 'left'}); display([p1, p2, p3, po, tp], scaling = constrained); end procnFig := 60;
Figs := seq(Fig(2*Pi*i/nFig), i = 0 .. nFig);
Error, (in Fig) invalid subscript selector
display(Figs, insequence = true);
NULL; Why this error message. Thank you.

## barycentre of 3 weighted points...

How to make this program more effective ?

A := [-3, 1, 2];
B := [-2, -1, 1];
C := [0, 3, -3];
A := [-3, 1, 2]

B := [-2, -1, 1]

C := [0, 3, -3]

alpha[1] := 2;#weight
alpha[2] := -1;
alpha[3] := 1;
alpha[1] := 2

alpha[2] := -1

alpha[3] := 1

x[1] := A[1];
x[2] := B[1];
x[3] := C[1];
x[1] := -3

x[2] := -2

x[3] := 0

y[1] := A[2];
y[2] := B[2];
y[3] := C[2];
y[1] := 1

y[2] := -1

y[3] := 3

z[1] := A[3];
z[2] := B[3];
z[3] := C[3];
z[1] := 2

z[2] := 1

z[3] := -3

sum(alpha[i], i = 1 .. 3);
2

xG := sum(alpha[i]*x[i], i = 1 .. 3)/sum(alpha[i], i = 1 .. 3);
xG := -2

yG := sum(alpha[i]*y[i], i = 1 .. 3)/sum(alpha[i], i = 1 .. 3);
yG := 3

zG := sum(alpha[i]*z[i], i = 1 .. 3)/sum(alpha[i], i = 1 .. 3);
zG := 0
Thank you.

## Error (in f) when datatype float[8]...

I dont know why I could not solve this problem.

I have attached my worksheet.

Please anyone help me to get solution to this problem.

Thank you so much

fypppp.mw

## How to run an executable program from Maple...

Dear All,

I have an executable program which I have generated with Fortran. Is it possible to run such a program from Maple? It sounds a bit weird but that would simplify my computation, since I would not need to use scripts in Linux. Thank you very much.

Cheers.

## Euler’s circle of an inscribed quadrilateral...

restart;
with(geometry):
with(plots):
_EnvHorizont:lName = 'x';
_EnvVerticalName = 'y';
Vdot := proc(U, V) local i; add(U[i]*V[i], i = 1 .. 2); end proc;
R := 5;
ang := [3/4*Pi, -(3*Pi)/4, -Pi/6,4*Pi/9];
seq(point(`||`(P, i), [R*cos(ang[i]), R*sin(ang[i])]), i = 1 .. 4);
pts:=[seq(P || i,i=1..4)]:
seq(dsegment(`||`(seg, i), [`||`(P, i), `||`(P, irem(i, 4) + 1)]), i = 1 .. 4);
Triangle(Tr1,[P1,P2,P4]);
EulerCircle(Elc1,Tr1,'centername'=o);
circle(cir, [point(OO, [0, 0]), R]);
dist := proc(M, N) sqrt(Vdot(M - N, M - N)); end proc;
display(draw([P1(color = black, symbol = solidcircle, symbolsize = 12),
P2(color = black, symbol = solidcircle, symbolsize = 12),
P3(color = black, symbol = solidcircle, symbolsize = 12),
P4(color = black, symbol = solidcircle, symbolsize = 12),seg1,
seg2,seg3,seq4,Tr1,Elc1,
cir(color = blue)]),
textplot([[seq( [ coordinates(`||`(P, i))[], convert(`||`(P, i), string)],i=1..4],
,align = [above, right]), axes = none);
does not recognize neg4 or tr1? I don't know to manage. Thank you.

## Difficulty with displat draw and textplot...

restart;
with(geometry):
with(plots):
_EnvHorizontalName = 'x':
_EnvVerticalName = 'y':
Vdot := proc(U, V) local i; add(U[i]*V[i], i = 1 .. 2); end proc:
R := 5:
ang := [2/3*Pi, -3*Pi*1/4, -Pi*1/6]:
seq(point(`||`(P, i), [R*cos(ang[i]), R*sin(ang[i])]), i = 1 .. 3):
seq(dsegment(`||`(seg, i), [`||`(P, i), `||`(P, irem(i, 3) + 1)]), i = 1 .. 3):
circle(cir, [point(OO, [0, 0]), R]):
dist := proc(M, N) sqrt(Vdot(M - N, M - N)); end proc:

display*([draw*[P1(color = black, symbol = solidcircle, symbolsize = 12),
P2(color = black, symbol = solidcircle, symbolsize = 12),
P3(color = black, symbol = solidcircle, symbolsize = 12),
cir(color = blue)],
textplot*([[coordinates(P1)[], "P1"],
[coordinates(P2)[], "P2"],
[coordinates(P3)[], "P3"]], align = [above, right])], axes = none);
/[
plots:-display |[geometry:-draw [
\[

P1(color = black, symbol = solidcircle, symbolsize = 12),

P2(color = black, symbol = solidcircle, symbolsize = 12),

P3(color = black, symbol = solidcircle, symbolsize = 12),

/[[-5  5  (1/2)      ]
cir(color = blue)], plots:-textplot |[[--, - 3     , "P1"],
\[[2   2             ]

[  5  (1/2)    5  (1/2)      ]  [5  (1/2)  -5      ]]
[- - 2     , - - 2     , "P2"], [- 3     , --, "P3"]],
[  2           2             ]  [2         2       ]]

\]             \
align = [above, right]|], axes = none|
/]             /

no figure drawn, Why? Thank you

## Given a function, I want maple to produce an ODE o...

I remember that once there was an option to provide to maple a given function and it produced a suitable DE that this function solves.

Can you remind me how to do it?

I remember there was such an option in the previous versions of maple.

## Projection of a point on a line...

do not accept substitution after line EQ
restart;
with(LinearAlgebra);
A := [1, -2];
B := [-2, 3];
C := [1, 1];
M := [x, y];
ProjPL := proc(C, A, B) local M, AB, AM, Q, eq, EQ, eq1, a, b, c, t, tt, n, dist, x, y, xH, yH, H, CH, no; M := [x, y]; AM := M - A; AB := B - A; Q := Matrix(2, [AM, AB]); eq := Determinant(Q); a := coeff(eq, x); b := coeff(eq, y); c := tcoeff(eq); dist := abs(a + b + c)/sqrt(a^2 + b^2); n := [a, b]; x := C[1] + n[1]*t; y := C[2] + n[2]*t; EQ := eq = 0; tt := solve(subs(x = C[1] + n[1]*t, y = C[2] + n[2]*t, EQ), t); xH := subs(t = tt, x); yH := subs(t = tt, y); H := [xH, yH]; CH := H - C; no := sqrt(CH[1]^2 + CH[2]^2); RETURN(EQ, dist, H, no); end proc;
ProjPL(C, A, B);

## Copying output together vs seperately...

So I have a problem when copying multiple output from Maple to Word. I am using a simple example here.

When I copy each part of the output as an image SEPERATELY, it comes out much cleaner than if I highlight everything together and then copy. Is there a reason this is the case and is there a way to fix this problem?

Here is the Maple Code:

with(plottools);
with(plots);
ttt1 := textplot([0, 1, "3  +  5  =  __", color = "black", font = ["Arial", "bold", 120]]);
display(ttt1, size = [1000, 200], axes = none);
ttt1 := textplot([0, 1, "1  +  3  =  __", color = "black", font = ["Arial", "bold", 120]]);
display(ttt1, size = [1000, 200], axes = none);

I also uploaded the Maple Worksheet

Here is the output:

There are TWO imges in the output in this example. I tried copying both together and pasting them, and then I tried copying them individually. The output is very different in quality.

The one copied alot is much cleaner once copied and pasted in Word. Does anyone why this is happening? I don't want to have to copy images individally for obvoius reasons. I know this example is only two images but what if there was alot more, etc.

Thank you

 >
 >
 >