Here is the corrected code:
 

Download BBM_new.mw

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
 

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

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

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);

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

Example of use:
dsx(0.1);

See this help page  rtable_indexing

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

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

Download BVP_new.mw

restart;
  rnge:=-1..1:
  fcns:=[[[0,0]]$5, x$10, [x,x^2]$10, [x,x^2,(1-x)]$10]:
  plots:-display( [ seq
                    ( plot
                      ( fcns[j], x=rnge),
                      j=1..35
                    )
                  ],
                  insequence=true, scaling=constrained
                );

Here is a more complicated option:

restart;
with(plots):
A:=animate(plot, [x, x=-1..a, color=red, thickness=2], a=-1..1):
B:=animate(plot, [x^2, x=-1..a, thickness=2, color=green], a=-1..1):
C:=animate(plot, [1-x, x=-1..a, color=blue, thickness=2], a=-1..1):
display([A, display(op([1,-1,1],A),B), display(op([1,-1,1],A),op([1,-1,1],B),C)], insequence, scaling=constrained);

                

See this post   https://mapleprimes.com/posts/207840-Combinations-Of-Multiple-Animations

Edit.

Download orthogonal-new1.mw

 I think the easiest way to create the same animation is the using a procedure that creates an one frame, and then using the plots:-animate command:

restart;

OneFrame:=proc(a)
local f, Point, Points1, Points2, Curve;
uses plots;
f:=x->x^2-x+2;
Point:=plot([[2,f(2)]],style=point, color=black, symbol=solidcircle, symbolsize=30);
Points1:=plot([[2-a,f(2-a)],[2+a,f(2+a)]], x=-3..5, style=point, color=red, symbol=solidbox, symbolsize=30);
Points2:=plot([[2-a,0],[2+a,0],[0,f(2-a)],[0,f(2+a)]], x=-3..5, style=point, color=red, symbol=diamond, symbolsize=30);
Curve:=plot(f, -3..5, 0..8, color=blue, thickness=3);
display(Point,Points1,Points2,Curve, gridlines, scaling=constrained);
end proc:

plots:-animate(OneFrame, [a], a=1..0.05);

See  https://en.wikipedia.org/wiki/MapleSim

This can be done in many ways. Here is a simple procedure with a for-loop that solves the problem:

P:=proc(A::Matrix, j::posint, a::numeric)
local i, m, n;
m,n:=upperbound(A);
for i from 1 to m do
if A[i,j]>a then return i fi;
od;
end proc:

Example:
A:=<1,2,1.1; 3,4,2.1; 5,6,3.1>;
P(A,3,2);

To draw this circle centered at point  H  and radius  R , you do not need to know any angles. Simply write the appropriate parametric equations and use  plot3d  command:

restart; 
with(geom3d):
 a := 3:
 b := 4:
 h := 5:
 point(A, 0, 0, 0):
 point(B, a, 0, 0):
 point(C, a, b, 0):
 point(DD, 0, b, 0):
 point(S, 0, 0, h):
 sphere(s, [A, B, C, S], [x,y,z], 'centername' = m); 
detail(s); 
plane(p, [S, A, B], [x, y, z]); 
n:=coordinates(projection(H, m, p)); 
R := distance(H, S);
S1:=plot3d(n+r*~([cos(t),0,sin(t)]), r=0..R, t=0..2*Pi, style=surface,  color=pink):
S2:=plots:-spacecurve(n+[R*cos(t),0,R*sin(t)], t=0..2*Pi, color="VioletRed", thickness=2):
S3:=plottools:-sphere([3/2, 2, 5/2], 5*sqrt(2)*(1/2), color="LightGreen", transparency=0.7):
plots:-display(S1,S2,S3, axes=normal, orientation=[20,70], lightmodel=light1);