Maple 2021 Questions and Posts

These are Posts and Questions associated with the product, Maple 2021

restart;
Fig:=proc(t)
local a,b,c,A,B,C,Oo,P,NorA,NorB,NorC,lieu,Lieu,dr,tx:
uses plots, geometry;
a := 11:b := 7:
c := sqrt(a^2 - b^2):

point(A, a*cos(t), b*sin(t)):
point(B, a*cos(t + 2/3*Pi), b*sin(t + 2/3*Pi)):
point(C, a*cos(t + 4/3*Pi), b*sin(t + 4/3*Pi)):
point(Oo,0,0):
lieu:=a^2*x^2+b^2*y^2-c^4/4=0:
Lieu := implicitplot(lieu, x = -a .. a, y = -b .. b, color = green):

line(NorA, y-coordinates(A)[2] =((a^2*coordinates(A)[2])/(b^2*coordinates(A)[1]))*(x-coordinates(A)[1]),[x, y]):
line(NorB, y-coordinates(B)[2] =((a^2*coordinates(B)[2])/(b^2*coordinates(B)[1]))*(x-coordinates(B)[1]), [x, y]):
line(NorC, y-coordinates(C)[2] =((a^2*coordinates(C)[2])/(b^2*coordinates(C)[1]))*(x-coordinates(C)[1]),[x, y]):
intersection(P,NorA,NorB):

ellipse(p, x^2/a^2 + y^2/b^2 - 1, [x, y]);

tx := textplot([[coordinates(A1)[], "A"],[coordinates(A2)[], "B"], [coordinates(C)[], "C"], [coordinates(Oo)[], "O"],#[coordinates(P)[], "P"]], font = [times, bold, 16], align = [above, left]):
dr := draw([p(color = blue),NorA(color=red,NorB(color=red),NorC(color=red),p(color=blue),
Oo(color = black, symbol = solidcircle, symbolsize = 8), P(color = black, symbol = solidcircle, symbolsize = 8)]):
display(dr,tx,Lieu,scaling=constrained, axes=none,title = "Les triangles inscrits dans une ellipse ont leur centre de gravité en son centre . Lieu du point de concours des perpendicalaires issues des sommets", titlefont = [HELVETICA, 14]);
end:

Error, `:=` unexpected
plots:-animate(Fig, [t], t=0.1..2*Pi, frames=150);
 

restart;
with(plots);
with(geometry);
_EnvHorizontalName := 'x';
_EnvVerticalName := 'y';
HC := HorizontalCoord;
VC := VerticalCoord;
a := 11;
b := 7;
t := (3*Pi)/8;
c := sqrt(a^2 - b^2);
ellipse(e1, x^2/a^2 + y^2/b^2 = 1);
point(Oo, 0, 0);
point(A, a*cos(t), b*sin(t));
point(B, a*cos(t + 2/3*Pi), b*sin(t + 2/3*Pi));
point(C, a*cos(t + 4/3*Pi), b*sin(t + 4/3*Pi));
point(G, (A[1] + B[1] + C[1])/3, (A[2] + B[2] + C[2])/3);
eval(coordinates(G));
line(NorA, y - A[2] = a^2*A[2]*(x - A[1])/b^2, [x, y]);
line(NorB, y - B[2] = a^2*B[2]*(x - B[1])/b^2, [x, y]);
line(NorC, y - C[2] = a^2*C[2]*(x - C[1])/b^2, [x, y]);
lieu := a^2*x^2 + b^2*y^2 - c^4/4 = 0;
Lieu := implicitplot(lieu, x = -a .. a, y = -b .. b, color = green);
tx := textplot([[coordinates(A)[], "A"], [coordinates(B)[], "B"], [coordinates(C)[], "C"], [coordinates(Oo)[], "O"]], font = [times, bold, 16], align = [above, left]);
dr := draw([e1(color = blue), NorA(color = red), NorB(color = red), NorC(color = red), A(color = red, symbol = solidcircle, symbolsize = 12), B(color = red, symbol = solidcircle, symbolsize = 12), C(color = red, symbol = solidcircle, symbolsize = 12), Oo(color = red, symbol = solidcircle, symbolsize = 12)]);
display([dr, tx, Lieu], scaling = constrained, axes = normal, title = "Ellipse et normales ", titlefont = [HELVETICA, 14]);
      [1        1        1       1        1        1     ]
      [- A[1] + - B[1] + - C[1], - A[2] + - B[2] + - C[2]]
      [3        3        3       3        3        3     ]

                              NorA

                              NorB

                              NorC

Warning, data could not be converted to float Matrix
Warning, data could not be converted to float Matrix
Warning, data could not be converted to float Matrix

On joint un point M  d'une ellipse aux foyers F1 et F2.  Les droites MF1 et MF2 recoupent l'ellipse aux points H1 et H2 ,  trouver l'enveloppe de la droite H1H2,  quand le point M se `déplace` sur l'ellipse.;


restart;
Fig := proc(t) local a, b, c, courbe, sol, sol1, dr, tx; _EnvHorizontalName := 'x'; _EnvVerticalName := 'y'; a := 11; b := 7; c := sqrt(a^2 - b^2); geometry:-ellipse(e1, x^2/a^2 + y^2/b^2 = 1); geometry:-point(Oo, 0, 0); geometry:-point(M, a*cos(t), b*sin(t)); geometry:-point(F1, -c, 0); geometry:-point(F2, c, 0); geometry:-line(MF1, [M, F1]); geometry:-line(MF2, [M, F2]); sol := solve({geometry:-Equation(MF1), x^2/a^2 + y^2/b^2 = 1}, {x, y}, explicit); print(%); geometry:-point(H1, subs(sol[2], x), subs(sol[2], y)); geometry:-line(MH1, [M, H1]); sol := solve({geometry:-Equation(MF2), x^2/a^2 + y^2/b^2 = 1}, {x, y}, explicit); print(%); geometry:-point(H2, subs(sol[2], x), subs(sol[2], y)); geometry:-line(MH2, [M, H2]); courbe := plots:-implicitplot(x^2/a^2 + (a^2 + c^2)^2*y^2/b^2 - 1 = 0, x = -a .. a, y = -b .. b, color = cyan); tx := plots:-textplot([[geometry:-coordinates(M)[], "M"], [geometry:-coordinates(Oo)[], "O"], [geometry:-coordinates(H1)[], "H1"], [geometry:-coordinates(H2)[], "H2"], [geometry:-coordinates(F1)[], "F1"], [geometry:-coordinates(F2)[], "F2"]], font = [times, bold, 16], align = [above, left]); dr := geometry:-draw([e1(color = blue), MH1(color = magenta), MH2(color = magenta), M(color = red, symbol = solidcircle, symbolsize = 12), H1(color = red, symbol = solidcircle, symbolsize = 12), H2(color = red, symbol = solidcircle, symbolsize = 12), F1(color = red, symbol = solidcircle, symbolsize = 12), F2(color = red, symbol = solidcircle, symbolsize = 12), Oo(color = red, symbol = solidcircle, symbolsize = 12)]); plots:-display([dr, tx, courbe], scaling = constrained, axes = normal, title = "Ellipse et normales ", titlefont = [HELVETICA, 14]); end proc;
Fig(Pi/3);
    /    11      7  (1/2)\    /      26411   210177  (1/2)  
   { x = --, y = - 3      }, { x = - ----- - ------ 2     , 
    \    2       2       /    \      57074   28537          

         11319  (1/2)  (1/2)   66199  (1/2)\ 
     y = ----- 3      2      - ----- 3      }
         28537                 57074       / 


     /    11      7  (1/2)\    /    210177  (1/2)   26411  
    { x = --, y = - 3      }, { x = ------ 2      - -----, 
     \    2       2       /    \    28537           57074  

            11319  (1/2)  (1/2)   66199  (1/2)\ 
      y = - ----- 3      2      - ----- 3      }
            28537                 57074       / 

Fig(Pi/6);
   /      104027  (1/2)   17787  (1/2)  (1/2)   123420  (1/2)
  { x = - ------ 3      + ----- 3      6      - ------ 2     
   \      22226           11113                 11113        

       19404  (1/2)  (1/2)        66199   11319  (1/2)\   
     + ----- 2      6     , y = - ----- + ----- 6      }, 
       11113                      22226   11113       /   

     /    11  (1/2)      7\ 
    { x = -- 3     , y = - }
     \    2              2/ 


Error, (in geometry:-line) the line is not defined
plots:-animate(Fig, [t], t = 0.1 .. 2*Pi, frames = 150);
            {x = -10.99908244, y = -0.09041172732}, 

              {x = 10.94504582, y = 0.6988339166}


Error, (in plots/animate) the line is not defined
;
NULL;
Thank you for your help.

intersections := proc(P, Q, T)
local R, W, w, t, a, b, sol, buff, v;
sol := []; if T = Y then W := X; else W := Y; end if;
R := resultant(P, Q, T);
print(`Résultant :`);
print(R);
w := fsolve(R, W); t := [];
for v in [w] do t := [op(t), fsolve(subs(W = v, P), T)]; end do;
for a in [w] do for b in [t] do if T = Y then buff := abs(subs(X = a, Y = b, P)) + abs(subs(X = a, Y = b, Q));
printf(`X=%a,   Y=%a   --->  %a\\n`, a, b, buff); if buff < 1/100000000 then sol := [op(sol), [a, b]]; end if;
else buff := abs(subs(X = b, Y = a, P)) + abs(subs(X = b, Y = a, Q));
printf(`X=%a,   Y=%a   --->  %a\\n`, b, a, buff); if buff < 1/100000000 then sol := [op(sol), [b, a]];
end if; end if; end do; end do;
printf(`Nombre de solutions :  %a\\n`, nops(sol)); print(sol); end proc;
Try with :
intersections(X^2 + Y^2 - 1, X - Y, X); Would you like to develop this procedure which does not give the number of solutions ?Thank you.

On considère une ellipse x^2/a^2+y^2/b^2-1=0 et 2 sommets de cette ellipse A(a,0) et B(0,b). On imagine une hyperbole équilatère variable passant par les points O, A et B. Cette courbe rencontre l'ellipse en 2 autres points A1 et B1. Montrer que la droite A1B1 passe par un point fixe. Même avec l'intelligence artificielle, je ne parviens pas à résoudre ce problème. Pourriez-vous d'aider. Merci.

Machine translation by moderator:

We consider an ellipse x^2/a^2+y^2/b^2-1=0 and 2 vertices of this ellipse A(a,0) and B(0,b). We imagine a variable equilateral hyperbola passing through the points O, A and B. This curve meets the ellipse at 2 other points A1 and B1. Show that the line A1B1 passes through a fixed point. Even with artificial intelligence, I can't solve this problem. Could you help. Thank you.

On donne une ellipse rapportée à ses axes x^2/a^2+y^2/b^2-1=0 et une droite (D) qui rencontre cette
courbe en 2 points A et B. 
On considère un cercle variable passant parles points A et B et on demande le lieu géométrique des points de rencontre des tangentes communes au cercle et à l'ellipse.
restart;
with(plots);
with(VectorCalculus);
a := 5;
b := 3;
ellipse_eq := (x, y) -> x^2/a^2 + y^2/b^2 - 1;
m := 1;
c := -2;
line_eq := (x, y) -> y - m*x - c;
intersections := solve({line_eq(x, y) = 0, ellipse_eq(x, y) = 0}, {x, y}, explicit);
A := intersections[1];
B := intersections[2];
A := [VectorCalculus:-`+`(VectorCalculus:-`*`(25, 17^VectorCalculus:-`-`(1)), VectorCalculus:-`*`(VectorCalculus:-`*`(15, sqrt(30)), 34^VectorCalculus:-`-`(1))), VectorCalculus:-`+`(VectorCalculus:-`-`(VectorCalculus:-`*`(9, 17^VectorCalculus:-`-`(1))), VectorCalculus:-`*`(VectorCalculus:-`*`(15, sqrt(30)), 34^VectorCalculus:-`-`(1)))];
B := [VectorCalculus:-`+`(VectorCalculus:-`*`(25, 17^VectorCalculus:-`-`(1)), VectorCalculus:-`-`(VectorCalculus:-`*`(VectorCalculus:-`*`(15, sqrt(30)), 34^VectorCalculus:-`-`(1)))), VectorCalculus:-`+`(VectorCalculus:-`-`(VectorCalculus:-`*`(9, 17^VectorCalculus:-`-`(1))), VectorCalculus:-`-`(VectorCalculus:-`*`(VectorCalculus:-`*`(15, sqrt(30)), 34^VectorCalculus:-`-`(1))))];
center_x := VectorCalculus:-`*`(VectorCalculus:-`+`(A[1], B[1]), 2^VectorCalculus:-`-`(1));
center_y := VectorCalculus:-`*`(VectorCalculus:-`+`(A[2], B[2]), 2^VectorCalculus:-`-`(1));
radius := VectorCalculus:-`*`(sqrt(VectorCalculus:-`+`(VectorCalculus:-`+`(A[1], VectorCalculus:-`-`(B[1]))^2, VectorCalculus:-`+`(A[2], VectorCalculus:-`-`(B[2]))^2)), 2^VectorCalculus:-`-`(1));
circle_eq := (x, y) -> (x - center_x)^2 + (y - center_y)^2 - radius^2;
L := (x1, y1, x2, y2, lambda1, lambda2) -> (x1 - x2)^2 + (y1 - y2)^2 + lambda1*ellipse_eq(x1, y1) + lambda2*circle_eq(x2, y2);
eq1 := diff(L(x1, y1, x2, y2, lambda1, lambda2), x1);
eq2 := diff(L(x1, y1, x2, y2, lambda1, lambda2), y1);
eq3 := diff(L(x1, y1, x2, y2, lambda1, lambda2), x2);
eq4 := diff(L(x1, y1, x2, y2, lambda1, lambda2), y2);
eq5 := ellipse_eq(x1, y1);
eq6 := circle_eq(x2, y2);
sols := solve({eq1, eq2, eq3, eq4, eq5, eq6}, {lambda1, lambda2, x1, x2, y1, y2}, explicit);
sols;
lieu_geometrique := [seq([sols[i][1], sols[i][2]], i = 1 .. nops(sols))];
plot(lieu_geometrique, style = point, symbol = cross, color = red, title = "Lieu géométrique des points de rencontre");
Ce code m'a été donné en partie par l'intelligence artificielle (Mistral), mais il se plante. Pourriez-vous corriger les erreurs. Merci.

On considère un cercle fixe O et un point fixe A extérieur. Une sécante variable BC à ce cercle passe par un point fixe J.
Démontrer que le cercle ABC passe par un second point fixe P.
restart;
Proc := proc(m)
local xA, yA, xB, yB, xC, yC, xJ, yJ, tx, dr, Oo, c1, r, eqBJ, eq1, sol;
_EnvHorizontalName := 'x'; _EnvVerticalName := 'y';
xJ := 5; yJ := 1; geometry:-point(A, 2, 4); geometry:-point(J, xJ, yJ); geometry:-point(Oo, 0, 0);
r := 3; c1 := plottools[geometry:-circle]([0, 0], r, color = blue);
eqBJ := y = m*(x - xJ) + yJ; geometry:-line(BJ, eqBJ, [x, y]);
eq1 := x^2 + y^2 = r^2; sol := solve({eqBJ, eq1}, {x, y}, explicit);
xB := subs(sol[1], x); yB := subs(sol[1], y);
geometry:-point(B, xB, yB); xC := subs(sol[2], x); yC := subs(sol[2], y);
geometry:-point(C, xC, yC); geometry:-circle(c2, [A, B, C]); geometry:-line(AB, [A, B]); geometry:-line(AC, [A, C]);
eqBJ := y = m*(x - xJ) + yJ; geometry:-line(BJ, eqBJ, [x, y]);
eq1 := x^2 + y^2 = r^2; sol := solve({eqBJ, eq1}, {x, y},explicit);
xB := subs(sol[1], x); yB := subs(sol[1], y); geometry:-point(B, xB, yB);
xC := subs(sol[2], x); yC := subs(sol[2], y); geometry:-point(C, xC, yC);
geometry:-circle(c2, [A, B, C]); geometry:-line(AB, [A, B]); geometry:-line(AC, [A, C]);
tx := plots:-textplot([[geometry:-coordinates(A)[], "A"], [geometry:-coordinates(B)[], "B"], [geometry:-coordinates(C)[], "C"], [geometry:-coordinates(J)[], "J"]], font = [times, bold, 16], align = [above, right]);
dr := geometry:-draw([AB(color = black), c2(color = magenta), A(color = blue, symbol = solidcircle, symbolsize = 16),
B(color = red, symbol = solidcircle, symbolsize = 16), C(color = red, symbol = solidcircle, symbolsize = 16),
J(color = red, symbol = solidcircle, symbolsize = 16)]); plots:-display([dr, c1, tx], axes = normal, view = [-5 .. 6, -4 .. 6], scaling = constrained);
end proc;
plots:-animate(Proc, [m], m = -0.9 .. 0.2*Pi, frames = 50);
Error, (in plots/animate) two lists or Vectors of numerical values expected
NULL;
I am trying to find out point P; Thank you for your help.

Here are the source codes for the paper  "Gaps Between Integers Having a Common Divisor with an Odd Semi-prime"


 

gg := proc (x, y) return abs(x-y)-1 end proc

proc (x, y) return abs(x-y)-1 end proc

(1)

CopyArrayElem := proc (x, n) local y, i; y := Array(1 .. n); for i to n do y(i) := x(i) end do; return y end proc

proc (x, n) local y, i; y := Array(1 .. n); for i to n do y(i) := x(i) end do; return y end proc

(2)

PrintArray := proc (x, n) local i; for i to n do printf("%d, ", x(i)); if i = (1/2)*n then printf("|,") end if end do; printf("\n") end proc

proc (x, n) local i; for i to n do printf("%d, ", x(i)); if i = (1/2)*n then printf("|,") end if end do; printf("
") end proc

(3)

GapArray := proc (x, n) local i, y; y := Array(1 .. n-1); for i to n-1 do y(i) := gg(x(i), x(i+1)) end do; return y end proc

proc (x, n) local i, y; y := Array(1 .. n-1); for i to n-1 do y(i) := gg(x(i), x(i+1)) end do; return y end proc

(4)

ShiftArrayElem := proc (x, n, d) local i, y; y := Array(1 .. n); for i to n do y(i) := x(i)+d end do; return y end proc

proc (x, n, d) local i, y; y := Array(1 .. n); for i to n do y(i) := x(i)+d end do; return y end proc

(5)

NULL

"chost:=proc(p,q) local i,j,m; local Hpq,hh;   Hpq:=Array(1..p+q-2);  hh:=Array(1..p+q-2);    for i from 1 to q-1 do   Hpq(i):=i*p;   od;      for j from 1 to p-1 do;   Hpq(i++):=j*q;   od;   hh:=sort(Hpq);    return hh;    end proc "

proc (p, q) local i, j, m, Hpq, hh; Hpq := Array(1 .. p+q-2); hh := Array(1 .. p+q-2); for i to q-1 do Hpq(i) := i*p end do; for j to p-1 do Hpq(`++`(i)) := j*q end do; hh := sort(Hpq); return hh end proc

(6)

nGroup := proc (p, m) local j, n, ar; ar := Array(1 .. m); for j from 0 to m-1 do n := ceil((j+1)*p/m)-floor(j*p/m)-1; ar(j+1) := n end do; return ar end proc

proc (p, m) local j, n, ar; ar := Array(1 .. m); for j from 0 to m-1 do n := ceil((j+1)*p/m)-floor(j*p/m)-1; ar(j+1) := n end do; return ar end proc

(7)

Position := proc (ary, k) local i, pos; pos := 0; for i to k-1 do pos := pos+ary(i) end do; pos := pos+1; return pos end proc

proc (ary, k) local i, pos; pos := 0; for i to k-1 do pos := pos+ary(i) end do; pos := pos+1; return pos end proc

(8)

groups := proc (p, q, lm, m) local k, j, x, r, ll, rr, ni, bl, br, i; i := 1; printf("______\n"); for k from 0 to m-2 do ll := floor(k*p/m); rr := floor((k+1)*p/m); ni := rr-ll; for j to ni do r := (j+ll)*m-k*p; printf("(%d: %d),", i, r); if j = ni then printf(" # %d p-hosts after q-host %d\n", lm+1, i*q) end if; i := i+1 end do; printf("\n") end do; ll := floor((m-1)*p/m); rr := p-1; ni := rr-ll; for j to ni do r := (j+ll)*m-k*p; printf("(%d: %d),", i, r); i := i+1 end do; printf("\n______\n") end proc

proc (p, q, lm, m) local k, j, x, r, ll, rr, ni, bl, br, i; i := 1; printf("______
"); for k from 0 to m-2 do ll := floor(k*p/m); rr := floor((k+1)*p/m); ni := rr-ll; for j to ni do r := (j+ll)*m-k*p; printf("(%d: %d),", i, r); if j = ni then printf(" # %d p-hosts after q-host %d
", lm+1, i*q) end if; i := i+1 end do; printf("
") end do; ll := floor((m-1)*p/m); rr := p-1; ni := rr-ll; for j to ni do r := (j+ll)*m-k*p; printf("(%d: %d),", i, r); i := i+1 end do; printf("
______
") end proc

(9)

idx := proc (p, m) local i, r, ri; for i to p-1 do r := i*m-floor(i*m/p)*p; if r = 1 or r = p-1 then ri := i end if; printf("%d, ", r) end do; printf("\n"); return ri end proc

proc (p, m) local i, r, ri; for i to p-1 do r := i*m-floor(i*m/p)*p; if r = 1 or r = p-1 then ri := i end if; printf("%d, ", r) end do; printf("
"); return ri end proc

(10)

DoTest := proc (p, q) local k, i, j, x, y, r, ni, lambda, n, g, xx, ll, rr, w, pos, nj, hh, grp; n := p+q-2; g := Array(1 .. n-1); hh := chost(p, q); printf("Hosts of p and q are:\n"); PrintArray(hh, n); lambda := floor(q/p); r := q-lambda*p; printf("Lambda=%d\nr=%d\n", lambda, r); printf("The %d elements in S(r,p) are:\n", p-1); for i to p-1 do printf("%d,", i*r) end do; printf("\n"); printf("The %d elements in rZ(r,p) are:\n", p-1); for j to p-1 do x := j*r-floor(j*r/p)*p; printf("%d,", x) end do; printf("\n"); printf("The %d subsets are as follows:\n", r); groups(p, q, lambda, r); printf("The maximum gap is: %d \n", p-1); g := q-p-1; printf("Total number of maximum gaps is:%d\n", g); g := GapArray(hh, n); xx := CopyArrayElem(hh, n-1); dataplot(xx, g) end proc

proc (p, q) local k, i, j, x, y, r, ni, lambda, n, g, xx, ll, rr, w, pos, nj, hh, grp; n := p+q-2; g := Array(1 .. n-1); hh := chost(p, q); printf("Hosts of p and q are:
"); PrintArray(hh, n); lambda := floor(q/p); r := q-lambda*p; printf("Lambda=%d
r=%d
", lambda, r); printf("The %d elements in S(r,p) are:
", p-1); for i to p-1 do printf("%d,", i*r) end do; printf("
"); printf("The %d elements in rZ(r,p) are:
", p-1); for j to p-1 do x := j*r-floor(j*r/p)*p; printf("%d,", x) end do; printf("
"); printf("The %d subsets are as follows:
", r); groups(p, q, lambda, r); printf("The maximum gap is: %d 
", p-1); g := q-p-1; printf("Total number of maximum gaps is:%d
", g); g := GapArray(hh, n); xx := CopyArrayElem(hh, n-1); dataplot(xx, g) end proc

(11)

``

DoTest(3, 5)

 

DoTest(5, 7)

 

 

 

 

NULL

DoTest(7, 9)

 

DoTest(11, 13)

 

DoTest(7, 11)

 

NULL

DoTest(5, 23)

 

DoTest(11, 47)

 

DoTest(13, 71)

 

DoTest(17, 29)

 

DoTest(23, 31)

 

DoTest(13, 23)

 

DoTest(11, 17)

 

 

DoTest(11, 29)

 

DoTest(13, 27)

 

DoTest(13, 79)

 

DoTest(11, 45)

 

DoTest(41, 71)

 

DoTest(47, 97)

 

DoTest(53, 103)

 

DoTest(101, 199)

 

DoTest(101, 205)

 

DoTest(23, 45)

 

DoTest(13, 25)

 

DoTest(13, 77)

 

DoTest(23, 93)

 

DoTest(23, 91)

 

DoTest(13, 25)

 

DoTest(13, 77)

 

DoTest(47, 91)

 

DoTest(53, 109)

 

NULL


 

Download Hosts.mw

How we can change identity like 1/sin(x)=csc(x) or 1/cos(x)=sec(x) sometime our function is beger than this and radical come in how i can do thus simplification?

restart

M := sin(x)/cos(x)

sin(x)/cos(x)

(1)

convert(M, trig)

sin(x)/cos(x)

(2)

tan(x)

tan(x)

(3)

simplify(M)

sin(x)/cos(x)

(4)

K := 1/sinh(x)

1/sinh(x)

(5)

simplify(convert(K, trig))

1/sinh(x)

(6)

csch(x)

csch(x)

(7)

Q := sqrt(beta[0]/(B[1]*cosh(xi*sqrt(-lambda))))

(beta[0]/(B[1]*cosh(xi*(-lambda)^(1/2))))^(1/2)

(8)

(beta[0]/(B[1]*cosh(xi*(-lambda)^(1/2))))^(1/2)

(9)

simplify((beta[0]/(B[1]*cosh(xi*(-lambda)^(1/2))))^(1/2), 'trig')

(beta[0]/(B[1]*cosh(xi*(-lambda)^(1/2))))^(1/2)

(10)
 

NULL

Download identity_change.mw

OneFrame := proc(k)
local Courbe, T, a, b, c, t, P, Q, NormM, F, Ell, sol, N1, N2, dr, tx;
a := 11; b := 7; c := sqrt(a^2 - b^2); t := 1/3*Pi;
Ell := x^2/a^2 + y^2/b^2 = 1;
geometry:-point(T, (a^2 - b^2)*cos(t)^3/a, -(a^2 - b^2)*sin(t)^3/b);
Courbe := plots:-implicitplot(Ell, x = -a - 10 .. a + 10, y = -b - 10 .. b + 10, scaling = constrained, color = blue);
NormM := plots:-implicitplot(y - b*sin(t) = a*sin(t)*(x - a*cos(t))/(b*cos(t)), x = -a - 5 .. a + 10, y = -b - 10 .. b + 10, color = orange); geometry:-line(Per, y - b*sin(t) = a*sin(t)*(x - a*cos(t))/(b*cos(t)), [x, y]);
geometry:-point(P, subs(y = 0
, geometry:-Equation(Per), 0));
geometry:-point(Q, 0, subs(x = 0, geometry:-Equation(Per)));
geometry:-point(M, a*cos(t), b*sin(t));
geometry:-point(N1, a*cos(k), b*sin(k));
geometry:-point(F, 2.329411765, -2.567510609);
geometry:-line(L, N1, F);
sol := solve({geometry:-Equation(L), Ell}, {x, y},explicit);
geometry:-point(N2, subs(sol[2], x), subs(sol[2], y));
geometry:-segment(sg, N1, N2);
tx := plots:-textplot([[geometry:-coordinates(M)[], "M"],
[geometry:-coordinates(N1)[], "N1"], [geometry:-coordinates(N2)[], "N2"],
[geometry:-coordinates(P)[], "P"],
[geometry:-coordinates(Q)[], "Q"],
[geometry:-coordinates(F)[], "F point de Frégier"],
[geometry:-coordinates(T)[], "T"]], font = [times, bold, 16], align = [above, left]);
dr := geometry:-draw([sg(color = magenta, linestyle = dash),
Per(color = black), P(color = red, symbol = solidcircle, symbolsize = 12),
Q(color = red, symbol = solidcircle, symbolsize = 12),
M(color = black, symbol = solidcircle, symbolsize = 12),
F(color = red, symbol = solidcircle, symbolsize = 12),
N1(color = black, symbol = solidcircle, symbolsize = 8),
N2(color = black, symbol = solidcircle, symbolsize = 8),
T(color = black, symbol = solidcircle, symbolsize = 8)]);
plots:-display(Courbe, tx, dr, scaling = constrained, axes = none); end proc;

plots:-animate(OneFrame, [k], k = Pi/3 .. Pi, frames = 50);
Error, (in plots/animate) wrong type of arguments
Why this animation does't work ? Thank you very much.
 

i use all code really sometime this is happen i can find each term seperatly without any factoring and collecting term i want all of them term by term i did all simplify code but i did't get result

restart

with(PDEtools)

NULL

with(Physics)

with(SolveTools)

undeclare(prime)

`There is no more prime differentiation variable; all derivatives will be displayed as indexed functions`

(1)

F := l((V(xi)^(1/(2*m)))^(4*m-2)*(V(xi)^(1/(2*m))*((1/4)*V(xi)^(1/(2*m))*(diff(V(xi), xi))^2/(m^2*V(xi)^2)+(1/2)*V(xi)^(1/(2*m))*(diff(diff(V(xi), xi), xi))/(m*V(xi))-(1/2)*V(xi)^(1/(2*m))*(diff(V(xi), xi))^2/(m*V(xi)^2))+(1/4)*(4*m+l-1)*(V(xi)^(1/(2*m)))^2*(diff(V(xi), xi))^2/(m^2*V(xi)^2)))*a+4*m((V(xi)^(1/(2*m)))^(4*m-2)*(V(xi)^(1/(2*m))*((1/4)*V(xi)^(1/(2*m))*(diff(V(xi), xi))^2/(m^2*V(xi)^2)+(1/2)*V(xi)^(1/(2*m))*(diff(diff(V(xi), xi), xi))/(m*V(xi))-(1/2)*V(xi)^(1/(2*m))*(diff(V(xi), xi))^2/(m*V(xi)^2))+(1/4)*(4*m+l-1)*(V(xi)^(1/(2*m)))^2*(diff(V(xi), xi))^2/(m^2*V(xi)^2)))*a+b[5]*(V(xi)^(1/(2*m)))^(8*m)+b[4]*(V(xi)^(1/(2*m)))^(6*m)+b[3]*(V(xi)^(1/(2*m)))^(6*m)+b[2]*(V(xi)^(1/(2*m)))^(4*m)+(1/2)*b[6]*(V(xi)^(1/(2*m)))^2*(diff(V(xi), xi))^2*(2*m-1)*(V(xi)^(1/(2*m)))^(-2+2*m)/(m*V(xi)^2)+2*(V(xi)^(1/(2*m)))^(2*m-1)*((1/4)*V(xi)^(1/(2*m))*(diff(V(xi), xi))^2/(m^2*V(xi)^2)+(1/2)*V(xi)^(1/(2*m))*(diff(diff(V(xi), xi), xi))/(m*V(xi))-(1/2)*V(xi)^(1/(2*m))*(diff(V(xi), xi))^2/(m*V(xi)^2))*m*b[6]+b[1]*(V(xi)^(1/(2*m)))^(2*m)-l*lambda = 0

l((V(xi)^((1/2)/m))^(4*m-2)*(V(xi)^((1/2)/m)*((1/4)*V(xi)^((1/2)/m)*(diff(V(xi), xi))^2/(m^2*V(xi)^2)+(1/2)*V(xi)^((1/2)/m)*(diff(diff(V(xi), xi), xi))/(m*V(xi))-(1/2)*V(xi)^((1/2)/m)*(diff(V(xi), xi))^2/(m*V(xi)^2))+(1/4)*(4*m+l-1)*(V(xi)^((1/2)/m))^2*(diff(V(xi), xi))^2/(m^2*V(xi)^2)))*a+4*m((V(xi)^((1/2)/m))^(4*m-2)*(V(xi)^((1/2)/m)*((1/4)*V(xi)^((1/2)/m)*(diff(V(xi), xi))^2/(m^2*V(xi)^2)+(1/2)*V(xi)^((1/2)/m)*(diff(diff(V(xi), xi), xi))/(m*V(xi))-(1/2)*V(xi)^((1/2)/m)*(diff(V(xi), xi))^2/(m*V(xi)^2))+(1/4)*(4*m+l-1)*(V(xi)^((1/2)/m))^2*(diff(V(xi), xi))^2/(m^2*V(xi)^2)))*a+b[5]*(V(xi)^((1/2)/m))^(8*m)+b[4]*(V(xi)^((1/2)/m))^(6*m)+b[3]*(V(xi)^((1/2)/m))^(6*m)+b[2]*(V(xi)^((1/2)/m))^(4*m)+(1/2)*b[6]*(V(xi)^((1/2)/m))^2*(diff(V(xi), xi))^2*(2*m-1)*(V(xi)^((1/2)/m))^(-2+2*m)/(m*V(xi)^2)+2*(V(xi)^((1/2)/m))^(2*m-1)*((1/4)*V(xi)^((1/2)/m)*(diff(V(xi), xi))^2/(m^2*V(xi)^2)+(1/2)*V(xi)^((1/2)/m)*(diff(diff(V(xi), xi), xi))/(m*V(xi))-(1/2)*V(xi)^((1/2)/m)*(diff(V(xi), xi))^2/(m*V(xi)^2))*m*b[6]+b[1]*(V(xi)^((1/2)/m))^(2*m)-l*lambda = 0

(2)

simplify(l((V(xi)^((1/2)/m))^(4*m-2)*(V(xi)^((1/2)/m)*((1/4)*V(xi)^((1/2)/m)*(diff(V(xi), xi))^2/(m^2*V(xi)^2)+(1/2)*V(xi)^((1/2)/m)*(diff(diff(V(xi), xi), xi))/(m*V(xi))-(1/2)*V(xi)^((1/2)/m)*(diff(V(xi), xi))^2/(m*V(xi)^2))+(1/4)*(4*m+l-1)*(V(xi)^((1/2)/m))^2*(diff(V(xi), xi))^2/(m^2*V(xi)^2)))*a+4*m((V(xi)^((1/2)/m))^(4*m-2)*(V(xi)^((1/2)/m)*((1/4)*V(xi)^((1/2)/m)*(diff(V(xi), xi))^2/(m^2*V(xi)^2)+(1/2)*V(xi)^((1/2)/m)*(diff(diff(V(xi), xi), xi))/(m*V(xi))-(1/2)*V(xi)^((1/2)/m)*(diff(V(xi), xi))^2/(m*V(xi)^2))+(1/4)*(4*m+l-1)*(V(xi)^((1/2)/m))^2*(diff(V(xi), xi))^2/(m^2*V(xi)^2)))*a+b[5]*(V(xi)^((1/2)/m))^(8*m)+b[4]*(V(xi)^((1/2)/m))^(6*m)+b[3]*(V(xi)^((1/2)/m))^(6*m)+b[2]*(V(xi)^((1/2)/m))^(4*m)+(1/2)*b[6]*(V(xi)^((1/2)/m))^2*(diff(V(xi), xi))^2*(2*m-1)*(V(xi)^((1/2)/m))^(-2+2*m)/(m*V(xi)^2)+2*(V(xi)^((1/2)/m))^(2*m-1)*((1/4)*V(xi)^((1/2)/m)*(diff(V(xi), xi))^2/(m^2*V(xi)^2)+(1/2)*V(xi)^((1/2)/m)*(diff(diff(V(xi), xi), xi))/(m*V(xi))-(1/2)*V(xi)^((1/2)/m)*(diff(V(xi), xi))^2/(m*V(xi)^2))*m*b[6]+b[1]*(V(xi)^((1/2)/m))^(2*m)-l*lambda = 0)

((V(xi)^((1/2)/m))^(4*m)*V(xi)*b[2]+((diff(diff(V(xi), xi), xi))*b[6]+V(xi)*b[1])*(V(xi)^((1/2)/m))^(2*m)+V(xi)*((b[3]+b[4])*(V(xi)^((1/2)/m))^(6*m)+l((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+4*m((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+b[5]*(V(xi)^((1/2)/m))^(8*m)-l*lambda))/V(xi) = 0

(3)

simplify(((V(xi)^((1/2)/m))^(4*m)*V(xi)*b[2]+((diff(diff(V(xi), xi), xi))*b[6]+V(xi)*b[1])*(V(xi)^((1/2)/m))^(2*m)+V(xi)*((b[3]+b[4])*(V(xi)^((1/2)/m))^(6*m)+l((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+4*m((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+b[5]*(V(xi)^((1/2)/m))^(8*m)-l*lambda))/V(xi) = 0)

((V(xi)^((1/2)/m))^(4*m)*V(xi)*b[2]+((diff(diff(V(xi), xi), xi))*b[6]+V(xi)*b[1])*(V(xi)^((1/2)/m))^(2*m)+V(xi)*((b[3]+b[4])*(V(xi)^((1/2)/m))^(6*m)+l((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+4*m((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+b[5]*(V(xi)^((1/2)/m))^(8*m)-l*lambda))/V(xi) = 0

(4)

numer(lhs(((V(xi)^((1/2)/m))^(4*m)*V(xi)*b[2]+((diff(diff(V(xi), xi), xi))*b[6]+V(xi)*b[1])*(V(xi)^((1/2)/m))^(2*m)+V(xi)*((b[3]+b[4])*(V(xi)^((1/2)/m))^(6*m)+l((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+4*m((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+b[5]*(V(xi)^((1/2)/m))^(8*m)-l*lambda))/V(xi) = 0))*denom(rhs(((V(xi)^((1/2)/m))^(4*m)*V(xi)*b[2]+((diff(diff(V(xi), xi), xi))*b[6]+V(xi)*b[1])*(V(xi)^((1/2)/m))^(2*m)+V(xi)*((b[3]+b[4])*(V(xi)^((1/2)/m))^(6*m)+l((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+4*m((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+b[5]*(V(xi)^((1/2)/m))^(8*m)-l*lambda))/V(xi) = 0)) = numer(rhs(((V(xi)^((1/2)/m))^(4*m)*V(xi)*b[2]+((diff(diff(V(xi), xi), xi))*b[6]+V(xi)*b[1])*(V(xi)^((1/2)/m))^(2*m)+V(xi)*((b[3]+b[4])*(V(xi)^((1/2)/m))^(6*m)+l((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+4*m((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+b[5]*(V(xi)^((1/2)/m))^(8*m)-l*lambda))/V(xi) = 0))*denom(lhs(((V(xi)^((1/2)/m))^(4*m)*V(xi)*b[2]+((diff(diff(V(xi), xi), xi))*b[6]+V(xi)*b[1])*(V(xi)^((1/2)/m))^(2*m)+V(xi)*((b[3]+b[4])*(V(xi)^((1/2)/m))^(6*m)+l((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+4*m((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+b[5]*(V(xi)^((1/2)/m))^(8*m)-l*lambda))/V(xi) = 0))

(V(xi)^((1/2)/m))^(4*m)*V(xi)*b[2]+(V(xi)^((1/2)/m))^(2*m)*V(xi)*b[1]+(V(xi)^((1/2)/m))^(6*m)*V(xi)*b[3]+(V(xi)^((1/2)/m))^(6*m)*V(xi)*b[4]+V(xi)*l((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+4*V(xi)*m((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+(V(xi)^((1/2)/m))^(8*m)*V(xi)*b[5]-V(xi)*l*lambda+(V(xi)^((1/2)/m))^(2*m)*(diff(diff(V(xi), xi), xi))*b[6] = 0

(5)

simplify((V(xi)^((1/2)/m))^(4*m)*V(xi)*b[2]+(V(xi)^((1/2)/m))^(2*m)*V(xi)*b[1]+(V(xi)^((1/2)/m))^(6*m)*V(xi)*b[3]+(V(xi)^((1/2)/m))^(6*m)*V(xi)*b[4]+V(xi)*l((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+4*V(xi)*m((1/4)*(V(xi)^((1/2)/m))^(4*m)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/(m^2*V(xi)^2))*a+(V(xi)^((1/2)/m))^(8*m)*V(xi)*b[5]-V(xi)*l*lambda+(V(xi)^((1/2)/m))^(2*m)*(diff(diff(V(xi), xi), xi))*b[6] = 0, 'symbolic')

V(xi)*(l((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/m^2)*a+4*m((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/m^2)*a+(diff(diff(V(xi), xi), xi))*b[6]+V(xi)^4*b[5]+(b[3]+b[4])*V(xi)^3+V(xi)^2*b[2]+V(xi)*b[1]-l*lambda) = 0

(6)

simplify(V(xi)*(l((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/m^2)*a+4*m((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/m^2)*a+(diff(diff(V(xi), xi), xi))*b[6]+V(xi)^4*b[5]+(b[3]+b[4])*V(xi)^3+V(xi)^2*b[2]+V(xi)*b[1]-l*lambda) = 0)

V(xi)*(l((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/m^2)*a+4*m((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/m^2)*a+(diff(diff(V(xi), xi), xi))*b[6]+V(xi)^4*b[5]+(b[3]+b[4])*V(xi)^3+V(xi)^2*b[2]+V(xi)*b[1]-l*lambda) = 0

(7)

normal(V(xi)*(l((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/m^2)*a+4*m((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/m^2)*a+(diff(diff(V(xi), xi), xi))*b[6]+V(xi)^4*b[5]+(b[3]+b[4])*V(xi)^3+V(xi)^2*b[2]+V(xi)*b[1]-l*lambda) = 0, ':-expanded')

V(xi)^3*b[2]+V(xi)^2*b[1]+V(xi)^4*b[3]+V(xi)^4*b[4]+V(xi)*l((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(diff(V(xi), xi))^2*l+2*(diff(V(xi), xi))^2*m)/m^2)*a+4*V(xi)*m((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(diff(V(xi), xi))^2*l+2*(diff(V(xi), xi))^2*m)/m^2)*a+V(xi)^5*b[5]-V(xi)*l*lambda+V(xi)*(diff(diff(V(xi), xi), xi))*b[6] = 0

(8)

normal(V(xi)^3*b[2]+V(xi)^2*b[1]+V(xi)^4*b[3]+V(xi)^4*b[4]+V(xi)*l((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(diff(V(xi), xi))^2*l+2*(diff(V(xi), xi))^2*m)/m^2)*a+4*V(xi)*m((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(diff(V(xi), xi))^2*l+2*(diff(V(xi), xi))^2*m)/m^2)*a+V(xi)^5*b[5]-V(xi)*l*lambda+V(xi)*(diff(diff(V(xi), xi), xi))*b[6] = 0)

V(xi)^3*b[2]+V(xi)^2*b[1]+V(xi)^4*b[3]+V(xi)^4*b[4]+V(xi)*l((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(diff(V(xi), xi))^2*l+2*(diff(V(xi), xi))^2*m)/m^2)*a+4*V(xi)*m((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(diff(V(xi), xi))^2*l+2*(diff(V(xi), xi))^2*m)/m^2)*a+V(xi)^5*b[5]-V(xi)*l*lambda+V(xi)*(diff(diff(V(xi), xi), xi))*b[6] = 0

(9)

eval(4*%*m^2)

(4*V(xi)^3*b[2]+4*V(xi)^2*b[1]+4*V(xi)^4*b[3]+4*V(xi)^4*b[4]+4*V(xi)*l((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(diff(V(xi), xi))^2*l+2*(diff(V(xi), xi))^2*m)/m^2)*a+16*V(xi)*m((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(diff(V(xi), xi))^2*l+2*(diff(V(xi), xi))^2*m)/m^2)*a+4*V(xi)^5*b[5]-4*V(xi)*l*lambda+4*V(xi)*(diff(diff(V(xi), xi), xi))*b[6])*m^2 = 0

(10)

simplify((4*V(xi)^3*b[2]+4*V(xi)^2*b[1]+4*V(xi)^4*b[3]+4*V(xi)^4*b[4]+4*V(xi)*l((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(diff(V(xi), xi))^2*l+2*(diff(V(xi), xi))^2*m)/m^2)*a+16*V(xi)*m((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(diff(V(xi), xi))^2*l+2*(diff(V(xi), xi))^2*m)/m^2)*a+4*V(xi)^5*b[5]-4*V(xi)*l*lambda+4*V(xi)*(diff(diff(V(xi), xi), xi))*b[6])*m^2 = 0)

4*V(xi)*m^2*(l((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/m^2)*a+4*m((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/m^2)*a+(diff(diff(V(xi), xi), xi))*b[6]+V(xi)^4*b[5]+(b[3]+b[4])*V(xi)^3+V(xi)^2*b[2]+V(xi)*b[1]-l*lambda) = 0

(11)

simplify(4*V(xi)*m^2*(l((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/m^2)*a+4*m((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/m^2)*a+(diff(diff(V(xi), xi), xi))*b[6]+V(xi)^4*b[5]+(b[3]+b[4])*V(xi)^3+V(xi)^2*b[2]+V(xi)*b[1]-l*lambda) = 0, 'symbolic')

4*V(xi)*m^2*(l((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/m^2)*a+4*m((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/m^2)*a+(diff(diff(V(xi), xi), xi))*b[6]+V(xi)^4*b[5]+(b[3]+b[4])*V(xi)^3+V(xi)^2*b[2]+V(xi)*b[1]-l*lambda) = 0

(12)

normal(4*V(xi)*m^2*(l((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/m^2)*a+4*m((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(l+2*m)*(diff(V(xi), xi))^2)/m^2)*a+(diff(diff(V(xi), xi), xi))*b[6]+V(xi)^4*b[5]+(b[3]+b[4])*V(xi)^3+V(xi)^2*b[2]+V(xi)*b[1]-l*lambda) = 0, ':-expanded')

4*V(xi)^5*m^2*b[5]+4*V(xi)^4*m^2*b[3]+4*V(xi)^4*m^2*b[4]+4*V(xi)^3*m^2*b[2]+4*V(xi)*m^2*(diff(diff(V(xi), xi), xi))*b[6]+4*V(xi)^2*m^2*b[1]+4*V(xi)*m^2*l((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(diff(V(xi), xi))^2*l+2*(diff(V(xi), xi))^2*m)/m^2)*a+16*V(xi)*m^2*m((1/4)*(2*(diff(diff(V(xi), xi), xi))*V(xi)*m+(diff(V(xi), xi))^2*l+2*(diff(V(xi), xi))^2*m)/m^2)*a-4*V(xi)*m^2*l*lambda = 0

(13)
 

NULL

Download simplify.mw

When i want reploting with the parameter i make them shorter by hand, make a problem for me and give me the same graph how fixed this problem? there is any code write in begind and give me all number about 2 decimal?

restart

K := [alpha = .33101604, theta = -2.54098361, mu = 4.89071038, k = 5.0, A[1] = 2.70491803, a = 3.63387978]

[alpha = .33101604, theta = -2.54098361, mu = 4.89071038, k = 5.0, A[1] = 2.70491803, a = 3.63387978]

(1)

MapleTA:-Builtin:-decimal(2, 20.8571)

20.86

(2)

MapleTA:-Builtin:-decimal(2, K)

Error, (in MapleTA:-Builtin:-decimal) invalid input: round expects its 1st argument, a1, to be of type algebraic, but received [100*(alpha = .33101604), 100*(theta = -2.54098361), 100*(mu = 4.89071038), 100*(k = 5.0), 100*(A[1] = 2.70491803), 100*(a = 3.63387978)]

 
 

NULL

Download decimal.mw

i found thus condition which if we substitute in equation must be equal to zero, i don't know  how i can get zero

test_pde1.mw

each time i use this i did not have any problem but this equation not seperate any one know what is problem?

restart

with(SolveTools)

undeclare(prime)

`There is no more prime differentiation variable; all derivatives will be displayed as indexed functions`

(1)

with(PDEtools)

P := U(xi)^3*mu*C[2]*h[9]+(2*I)*(diff(U(xi), xi))*a*k*mu+4*(diff(U(xi), xi))*k*mu^3*C[2]*h[7]-4*(diff(diff(diff(U(xi), xi), xi), xi))*k^3*mu*C[2]*h[7]-U(xi)^3*mu*C[2]*h[8]+I*(diff(diff(diff(diff(U(xi), xi), xi), xi), xi))*k^4*C[2]*h[7]+I*(diff(U(xi), xi))*U(xi)^2*k*C[2]*h[9]-(6*I)*(diff(diff(U(xi), xi), xi))*k^2*mu^2*C[2]*h[7]+I*U(xi)*mu^4*C[2]*h[7]-I*(diff(U(xi), xi))*v-U(xi)*w+b*U(xi)^3-U(xi)*a*mu^2+(diff(diff(U(xi), xi), xi))*a*k^2+I*(diff(U(xi), xi))*U(xi)^2*k*C[2]*h[8]+C[1](-U(xi)^3*mu^2*h[2]+(2*I)*(diff(U(xi), xi))*U(xi)^2*k*mu*h[4]-(2*I)*(diff(U(xi), xi))*U(xi)^2*k*mu*h[5]+(diff(U(xi), xi))^2*U(xi)*k^2*h[2]-U(xi)^3*mu^2*h[5]+U(xi)^2*(diff(diff(U(xi), xi), xi))*k^2*h[5]-(4*(diff(U(xi), xi))*I)*k*mu^3*h[1]+4*(diff(diff(diff(U(xi), xi), xi), xi))*k^3*mu*h[1]*I+(2*I)*(diff(U(xi), xi))*U(xi)^2*k*mu*h[2]+h[6]*U(xi)^5-U(xi)^3*mu^2*h[4]+U(xi)^2*(diff(diff(U(xi), xi), xi))*k^2*h[4]+U(xi)*mu^4*h[1]-6*(diff(diff(U(xi), xi), xi))*k^2*mu^2*h[1]+(diff(diff(diff(diff(U(xi), xi), xi), xi), xi))*k^4*h[1]+h[3](k^2*(diff(U(xi), xi))^2+2*(0+I)*(diff(U(xi), xi))*k*mu*U(xi)-mu^2*U(xi)^2)*U(xi)) = 0

U(xi)^3*mu*C[2]*h[9]+I*(diff(U(xi), xi))*U(xi)^2*k*C[2]*h[8]+4*(diff(U(xi), xi))*k*mu^3*C[2]*h[7]-4*(diff(diff(diff(U(xi), xi), xi), xi))*k^3*mu*C[2]*h[7]-U(xi)^3*mu*C[2]*h[8]+I*(diff(U(xi), xi))*U(xi)^2*k*C[2]*h[9]-(6*I)*(diff(diff(U(xi), xi), xi))*k^2*mu^2*C[2]*h[7]+I*U(xi)*mu^4*C[2]*h[7]-I*(diff(U(xi), xi))*v+(2*I)*(diff(U(xi), xi))*a*k*mu-U(xi)*w+b*U(xi)^3-U(xi)*a*mu^2+(diff(diff(U(xi), xi), xi))*a*k^2+I*(diff(diff(diff(diff(U(xi), xi), xi), xi), xi))*k^4*C[2]*h[7]+C[1](-U(xi)^3*mu^2*h[2]+(2*I)*(diff(U(xi), xi))*U(xi)^2*k*mu*h[4]-(4*I)*(diff(U(xi), xi))*k*mu^3*h[1]+(diff(U(xi), xi))^2*U(xi)*k^2*h[2]-U(xi)^3*mu^2*h[5]+U(xi)^2*(diff(diff(U(xi), xi), xi))*k^2*h[5]+(4*I)*(diff(diff(diff(U(xi), xi), xi), xi))*k^3*mu*h[1]-(2*I)*(diff(U(xi), xi))*U(xi)^2*k*mu*h[5]+(2*I)*(diff(U(xi), xi))*U(xi)^2*k*mu*h[2]+h[6]*U(xi)^5-U(xi)^3*mu^2*h[4]+U(xi)^2*(diff(diff(U(xi), xi), xi))*k^2*h[4]+U(xi)*mu^4*h[1]-6*(diff(diff(U(xi), xi), xi))*k^2*mu^2*h[1]+(diff(diff(diff(diff(U(xi), xi), xi), xi), xi))*k^4*h[1]+h[3](k^2*(diff(U(xi), xi))^2+(2*I)*(diff(U(xi), xi))*k*mu*U(xi)-mu^2*U(xi)^2)*U(xi)) = 0

(2)

Re(P)

Re(U(xi)^3*mu*C[2]*h[9]+4*(diff(U(xi), xi))*k*mu^3*C[2]*h[7]-4*(diff(diff(diff(U(xi), xi), xi), xi))*k^3*mu*C[2]*h[7]-U(xi)^3*mu*C[2]*h[8]-U(xi)*w+b*U(xi)^3-U(xi)*a*mu^2+(diff(diff(U(xi), xi), xi))*a*k^2+C[1](-U(xi)^3*mu^2*h[2]+(2*I)*(diff(U(xi), xi))*U(xi)^2*k*mu*h[4]-(4*I)*(diff(U(xi), xi))*k*mu^3*h[1]+(diff(U(xi), xi))^2*U(xi)*k^2*h[2]-U(xi)^3*mu^2*h[5]+U(xi)^2*(diff(diff(U(xi), xi), xi))*k^2*h[5]+(4*I)*(diff(diff(diff(U(xi), xi), xi), xi))*k^3*mu*h[1]-(2*I)*(diff(U(xi), xi))*U(xi)^2*k*mu*h[5]+(2*I)*(diff(U(xi), xi))*U(xi)^2*k*mu*h[2]+h[6]*U(xi)^5-U(xi)^3*mu^2*h[4]+U(xi)^2*(diff(diff(U(xi), xi), xi))*k^2*h[4]+U(xi)*mu^4*h[1]-6*(diff(diff(U(xi), xi), xi))*k^2*mu^2*h[1]+(diff(diff(diff(diff(U(xi), xi), xi), xi), xi))*k^4*h[1]+h[3](k^2*(diff(U(xi), xi))^2+(2*I)*(diff(U(xi), xi))*k*mu*U(xi)-mu^2*U(xi)^2)*U(xi)))-Im((diff(U(xi), xi))*U(xi)^2*k*C[2]*h[8]+(diff(U(xi), xi))*U(xi)^2*k*C[2]*h[9]-6*(diff(diff(U(xi), xi), xi))*k^2*mu^2*C[2]*h[7]+U(xi)*mu^4*C[2]*h[7]-(diff(U(xi), xi))*v+2*(diff(U(xi), xi))*a*k*mu+(diff(diff(diff(diff(U(xi), xi), xi), xi), xi))*k^4*C[2]*h[7]) = 0

(3)
 

``

Download real_and_imaginary_.mw

I have  a big problem in transformation How we can do suh transformation in  type of  procure  without use any hand work for example in physic abs|-| remove the exponential term how the maple remove that term automatically and collect all term and do my transformation this example is really hard one which is must do a lot by hand and mixed them which maybe a week take my time to get results and how i reach the results without spending that time i have a result of this equation and i am try to get but i don't know the results of this person is correct or not but i will share in here,  i did some try i will share in here too if in DEchange add U(xi) it will work and give me the other step but i need something more effective, when q^* is conjugate of q =exp(-ipsi(x,t))U(xi)

NULL

restart

with(PDEtools)

with(Physics)

with(SolveTools)

undeclare(prime)

`There is no more prime differentiation variable; all derivatives will be displayed as indexed functions`

(1)

 

 

tr := {t = tau, x = xi/k+v*tau^alpha/(k*alpha)+theta, u(x, t) = U(xi)*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta)), u[1](x, t) = U(xi)*exp(-I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))}

{t = tau, x = xi/k+v*tau^alpha/(k*alpha)+theta, u(x, t) = U(xi)*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta)), u[1](x, t) = U(xi)*exp(-I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))}

(2)

pde := I*(I*U(xi)*exp(I*(xi/k+v*tau^alpha/(k*alpha)-mu*tau+theta))*w-exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(U(xi), xi))*v)+a*(diff(u(x, t), `$`(x, 2)))+b*U(xi)^2*u(x, t)+C[1](h[1]*(diff(u(x, t), `$`(x, 4)))+h[2]*(diff(u(x, t), x))^2*u[1](x, t)+h[3]*abs(diff(u(x, t), x))^2*u(x, t)+h[4]*U(xi)^2*(diff(u(x, t), `$`(x, 2)))+h[5]*u(x, t)^2*(diff(u[1](x, t), `$`(x, 2)))+h[6]*U(xi)^4*u(x, t))+I*C[2]*(h[7]*(diff(u(x, t), `$`(x, 4)))+h[8]*U(xi)^2*(diff(u(x, t), x))+h[9]*u(x, t)^2*(diff(u[1](x, t), x))) = 0

I*(I*U(xi)*exp(I*(xi/k+v*tau^alpha/(k*alpha)-mu*tau+theta))*w-exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(U(xi), xi))*v)+a*(diff(diff(u(x, t), x), x))+b*U(xi)^2*u(x, t)+C[1](h[1]*(diff(diff(diff(diff(u(x, t), x), x), x), x))+h[2]*(diff(u(x, t), x))^2*u[1](x, t)+h[3]*abs(diff(u(x, t), x))^2*u(x, t)+h[4]*U(xi)^2*(diff(diff(u(x, t), x), x))+h[5]*u(x, t)^2*(diff(diff(u[1](x, t), x), x))+h[6]*U(xi)^4*u(x, t))+I*C[2]*(h[7]*(diff(diff(diff(diff(u(x, t), x), x), x), x))+h[8]*U(xi)^2*(diff(u(x, t), x))+h[9]*u(x, t)^2*(diff(u[1](x, t), x))) = 0

(3)

``

PDEtools:-dchange(tr, pde, [xi, tau, U, U(xi)])

I*(I*U(xi)*exp(I*(xi/k+v*tau^alpha/(k*alpha)-mu*tau+theta))*w-exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(U(xi), xi))*v)+a*((2*I)*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(U(xi), xi))/k+exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(diff(U(xi), xi), xi))-U(xi)*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))/k^2)*k^2+b*U(xi)^3*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))+C[1](h[1]*(-(4*I)*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(U(xi), xi))/k^3-6*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(diff(U(xi), xi), xi))/k^2+(4*I)*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(diff(diff(U(xi), xi), xi), xi))/k+exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(diff(diff(diff(U(xi), xi), xi), xi), xi))+U(xi)*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))/k^4)*k^4+h[2]*(exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(U(xi), xi))+I*U(xi)*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))/k)^2*k^2*U(xi)*exp(-I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))+h[3]*abs((exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(U(xi), xi))+I*U(xi)*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))/k)*k)^2*U(xi)*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))+h[4]*U(xi)^2*((2*I)*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(U(xi), xi))/k+exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(diff(U(xi), xi), xi))-U(xi)*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))/k^2)*k^2+h[5]*U(xi)^2*(exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta)))^2*((diff(diff(U(xi), xi), xi))*exp(-I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))-(2*I)*(diff(U(xi), xi))*exp(-I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))/k-U(xi)*exp(-I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))/k^2)*k^2+h[6]*U(xi)^5*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta)))+I*C[2]*(h[7]*(-(4*I)*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(U(xi), xi))/k^3-6*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(diff(U(xi), xi), xi))/k^2+(4*I)*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(diff(diff(U(xi), xi), xi), xi))/k+exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(diff(diff(diff(U(xi), xi), xi), xi), xi))+U(xi)*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))/k^4)*k^4+h[8]*U(xi)^2*(exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))*(diff(U(xi), xi))+I*U(xi)*exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))/k)*k+h[9]*U(xi)^2*(exp(I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta)))^2*((diff(U(xi), xi))*exp(-I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))-I*U(xi)*exp(-I*(xi/k+v*tau^alpha/(k*alpha)+mu*tau+theta))/k)*k) = 0

(4)
 

NULL


Download find_ODE.mw

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