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Kitonum

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Two simple ways...

Kitonum 21435
October 15 2019
0 1

Here are 2 easy ways to do this without a long fraction bar:

restart;
(1/x)*(x^3-2*x^2+5*x-7);  # With long fraction bar
``(1/x)*(x^3-2*x^2+5*x-7);  # The first way
(x)^``(-1)*(x^3-2*x^2+5*x-7);  # The second way

                           

 

All the possibilities...

Kitonum 21435
October 15 2019
1 1

Here are all the $ 200 shopping options, sorted by increasing Gb:

sort([seq(seq(seq([s=32*x+64*y+128*z,['x'=x,'y'=y,'z'=z]],z=floor((200-15*x-20*y)/30)),y=0..floor((200-15*x)/20)),x=0..floor(200/15))], key=(p->rhs(p[1]))):
select(p->floor((200-eval(15*x + 20*y + 30*z,p[2]))/15)<1 and floor((200-eval(15*x + 20*y + 30*z,p[2]))/20)<1 and floor((200-eval(15*x + 20*y + 30*z,p[2]))/30)<1, %)[];
nops([%]); # The number of all the possibilities

       [s = 416, [x = 13, y = 0, z = 0]], [s = 448, [x = 10, y = 2, z = 0]], [s = 448, [x = 12, y = 1, z = 0]], [s = 480, [x = 9, y = 3, z = 0]], [s = 480, [x = 11, y = 0, z = 1]], [s = 512, [x = 6, y = 5, z = 0]], [s = 512, [x = 8, y = 2, z = 1]], [s = 512, [x = 8, y = 4, z = 0]], [s = 512, [x = 10, y = 1, z = 1]], [s = 544, [x = 5, y = 6, z = 0]], [s = 544, [x = 7, y = 3, z = 1]], [s = 544, [x = 9, y = 0, z = 2]], [s = 576, [x = 2, y = 8, z = 0]], [s = 576, [x = 4, y = 5, z = 1]], [s = 576, [x = 4, y = 7, z = 0]], [s = 576, [x = 6, y = 2, z = 2]], [s = 576, [x = 6, y = 4, z = 1]], [s = 576, [x = 8, y = 1, z = 2]], [s = 608, [x = 1, y = 9, z = 0]], [s = 608, [x = 3, y = 6, z = 1]], [s = 608, [x = 5, y = 3, z = 2]], [s = 608, [x = 7, y = 0, z = 3]], [s = 640, [x = 0, y = 8, z = 1]], [s = 640, [x = 0, y = 10, z = 0]], [s = 640, [x = 2, y = 5, z = 2]], [s = 640, [x = 2, y = 7, z = 1]], [s = 640, [x = 4, y = 2, z = 3]], [s = 640, [x = 4, y = 4, z = 2]], [s = 640, [x = 6, y = 1, z = 3]], [s = 672, [x = 1, y = 6, z = 2]], [s = 672, [x = 3, y = 3, z = 3]], [s = 672, [x = 5, y = 0, z = 4]], [s = 704, [x = 0, y = 5, z = 3]], [s = 704, [x = 0, y = 7, z = 2]], [s = 704, [x = 2, y = 2, z = 4]], [s = 704, [x = 2, y = 4, z = 3]], [s = 704, [x = 4, y = 1, z = 4]], [s = 736, [x = 1, y = 3, z = 4]], [s = 736, [x = 3, y = 0, z = 5]], [s = 768, [x = 0, y = 2, z = 5]], [s = 768, [x = 0, y = 4, z = 4]], [s = 768, [x = 2, y = 1, z = 5]], [s = 800, [x = 1, y = 0, z = 6]], [s = 832, [x = 0, y = 1, z = 6]]
                                                                            44


We see that  s=832  can only be obtained in one way.

 

Edit.
 

Nested seq...

Kitonum 21435
October 14 2019
0 0 
[seq(seq(`if`(5*x + 3*y = 100, [x,y], NULL), x=0..20), y=0..20)];
plots:-pointplot(%);

 

Euclid's algorithms...

Kitonum 21435
October 14 2019
1 0

Here are the Euclidean algorithm and the extended Euclidean algorithm.

Euclidean algorithm 

 

> 

GCD:=proc(a,b)
local r, r1, b1;
r:=a-b*iquo(a,b); b1:=b;
while r>0 do
r1:=b1-r*iquo(b1,r); b1:=r; r:=r1;
od;
b1;
end proc:
> 

GCD(2680,3244);  # Example

4

 (1)
> 

 

Extended Euclidean algorithm
 

> 

ExtendedEuclid:=proc(a::nonnegint,b::nonnegint)
local d, x, y, x1, y1, x2, y2, a1, b1, q, r;
if a<b then error "Should be a>=b" fi;
if b=0 then d:=a; x:=1; y:=0; return [d,x,y] fi;
x2:=1; x1:=0; y2:=0; y1:=1; a1:=a; b1:=b;
while b1>0 do
q:=floor(a1/b1); r:=a1-q*b1; x:=x2-q*x1; y:=y2-q*y1;
a1:=b1; b1:=r; x2:=x1; x1:=x; y2:=y1; y1:=y;
od;
d:=a1; x:=x2; y:=y2;
[d,x,y];
end proc:
> 

ExtendedEuclid(30,12);  # Example

[6, 1, -2]

 (2)
> 

30*1+12*(-2);  # Check

6

 (3)
> 

ExtendedEuclid(65208, 18344);  # Your example

[8, 539, -1916]

 (4)
> 

65208*539+18344*(-1916);  # Check

8

 (5)
> 

 


 

Download Euclid.mw

Plotting of vectors...

Kitonum 21435
October 14 2019
1 3

You can use the commands  plots:-arrow, plots:-display, LinearAlgebra:-CrossProduct  to plot vectors, the results of operations on them, and animations of these. See examples below:


 

> 

restart;
with(plots):
with(LinearAlgebra):
> 

u, v, w:=<1, 2, 3>, <-1, 2, 3>, <-1, 2, -2>:
U:=arrow(u, color=blue):
V:=arrow(u, v, color=blue):
W:=arrow(u+v, w, color=blue):
> 

r1:=u+v+w: r2:=u-v: r3:=CrossProduct(u,v):
R1:=arrow(r1, color=red):
R2:=arrow(v, r2, color=red):
R3:=arrow(CrossProduct(u,v), color=red):
display(U, V, W, R1, scaling=constrained);
V1:=arrow(v, color=blue):
display(U, V1, R2, scaling=constrained);
display(U, V1, R3, scaling=constrained);

 

 

 
> 

display(plot3d([[0,0,0]])$5, display(U)$10, display([U,V])$10, display([U,V,W])$10, display([U,V,W,R1])$10, insequence, scaling=constrained);

 
> 

 


 

Download Plotting_Vectors.mw

Explore...

Kitonum 21435
October 11 2019
0 2

Try the  Explore  command for this:


 

> 

restart;
f:=1+B*x-(1/12)*B*x^3+0.1666666667e-4*B^3*x^3-4.166666667*10^(-8)*B^4*x^4+(1/160)*B*x^5+8.333333333*10^(-11)*B^5*x^5-0.5000000000e-2*B^2*x^2+0.1666666667e-4*B*x^3*C^2-4.166666667*10^(-8)*B*x^4*C^3+8.333333333*10^(-11)*B*x^5*C^4-0.5000000000e-2*B*C*x^2+0.3333333333e-4*B^2*x^3*C-1.250000000*10^(-7)*B^3*x^4*C-1.250000000*10^(-7)*B^2*x^4*C^2+3.333333333*10^(-10)*B^4*x^5*C+5.000000000*10^(-10)*B^3*x^5*C^2+3.333333333*10^(-10)*B^2*x^5*C^3+0.7291666667e-3*B*x^4*C-0.3333333333e-5*B*x^5*C^2+0.6250000000e-3*B^2*x^4-0.2083333333e-5*B^3*x^5-0.5416666667e-5*B^2*x^5*C:
Explore(plot(f, x=-4..4, view=-10..10), B=-5...5., C=-5...5.);
> 

 



It can be noted that changing the parameter  C  has very little effect on the graph

 

Download 2param.mw

Syntax...

Kitonum 21435
October 09 2019
2 5

Here is the corrected code:
 

> 

restart:
with(PDEtools):
PDE :=  diff(y(x,t), t)-diff(y(x,t), x,x,t)-diff(y(x,t), x$2)+ diff(y(x,t), x)+y(x,t)*diff(y(x,t),x)=exp(-t)*(cos(x)-sin(x)+1/2*exp(-t)*sin(2*x));

diff(y(x, t), t)-(diff(diff(diff(y(x, t), t), x), x))-(diff(diff(y(x, t), x), x))+diff(y(x, t), x)+y(x, t)*(diff(y(x, t), x)) = exp(-t)*(cos(x)-sin(x)+(1/2)*exp(-t)*sin(2*x))

 (1)
> 

# Initial/boundary conditions
  BCs:=y(0,t) = 0, y(Pi,t)=0;
  ICs:=y(x,0) =sin(x) ;

y(0, t) = 0, y(Pi, t) = 0

  

y(x, 0) = sin(x)

 (2)
> 

pdsolve({PDE, BCs,ICs}, y(x,t));  # NULL. Maple does not find any exact solution
> 

num_sol := pdsolve(PDE, {BCs,ICs}, numeric);
num_sol:-plot3d(x=0..1, t=0..1);

_m2208546638688

  

 
> 

 

> 

 
 exact_solution:=exp(-t)*sin(x);
Test1:=pdetest(y(x,t)=exact_solution,[PDE, BCs,ICs]);

exp(-t)*sin(x)

  

[0, 0, 0, 0]

 (3)
> 

 


 

Download BBM_new.mw

The simple recursive procedure...

Kitonum 21435
October 07 2019
1 2 
restart;
P:=proc(n)
option remember;
P(0):=0; P(1):=1;
if n>=2 and P(n-1)-1 in [seq(P(k),k=0..n-2)] then return 2*P(n-1) else P(n-1)-1 fi;
end proc:


Examples of use:

P(100);
seq('P(k)', k=0..50);             

                                                                121
0, 1, 2, 4, 3, 6, 5, 10, 9, 8, 7, 14, 13, 12, 11, 22, 21, 20, 19,  18, 17, 16, 15, 30, 29, 28, 27, 26, 25, 24, 23, 46, 45, 44, 43,   42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 62, 61, 60, 59
 

Student:-Calculus1:-Asymptotes...

Kitonum 21435
October 07 2019
1 0

In the previous answer, the asymptote  y=-x  is omitted. To automatically find all asymptotes (inclined and vertical), use  Student:-Calculus1:-Asymptotes  command:

y:=sqrt(1+x^2):
Student:-Calculus1:-Asymptotes(y,x);
plot([y,x,-x], x=-5..5, -1..5, color=[red,black$2], thickness=[2,1$2], linestyle=[1,3$2], scaling=constrained); # The plotting for a visualization

                  

 

 

Numerical solutions...

Kitonum 21435
October 06 2019
1 5

It seems that the natural parameterization and calculation of the arc length in this example can only be done numerically:

restart;
r:= t-><cos(2*t), sin(3*t), 4*t>:
L:=unapply(int(sqrt(add(diff(r(t),t)^~2)), t=0..t),t);
f:=s->fsolve(s=L(t),t=0..infinity);
f(1);  # Example
g:=s->r(f(s));
g(1);  # Example

L:=unapply(int(sqrt(add(diff(r(t),t)^~2)), t=t1..t2),t1,t2);
evalf(L(1,3));  # Example

                  

 

Solution...

Kitonum 21435
October 05 2019
0 2 
restart;
a:= plots:-display(plottools :- arrow( [1, -2], [4, -1], 0.001, 0.05, 0.1)):
b:= plots:-display(plottools :- arrow( [-1, 3], [2, 1], 0.001, 0.05, 0.1)):
c:= plots:-display(plottools :- arrow( [0, 2], [-3, 0], 0.001, 0.05, 0.1)):
plots:-display([plot([[0,0]])$5,a$10,plots:-display([a,b])$10,plots:-display([a,b,c])$10], insequence);

     

 

Syntax...

Kitonum 21435
October 03 2019
2 1

Replace the last line with
dsx := dsolve(sys_ode, numeric);

Example of use:
dsx(0.1);

Indexing Arrays, Matrices, and Vectors i...

Kitonum 21435
October 02 2019
1 0

See this help page  rtable_indexing

Other initial conditions...

Kitonum 21435
October 01 2019
2 5

From Fig. 2 it is clearly seen that  f(0)=1  and  D(f)(0)<0 , which contradicts your initial conditions.

Explore...

Kitonum 21435
October 01 2019
1 0

 Jayaprakash J   For movable boundary conditions you can use the  Explore  command.

 

Download BVP_new.mw

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