Unfortunately, Maple doesn't think to simplify the sum of the first and third terms after the simplification, but we can tell him to do so:

Download simplification.mw

If we want the number  n  to match the number of petals, then

restart;
Rose:= (n::integer)-> plots:-polarplot(`if`(n::even,sin(n/2*t),sin(n*t)), filled, color= `if`(n::odd, pink, aquamarine)):

Rose(4);  Rose(3); # Examples

@mylikes  You didn't specify your version of Maple. It's probably a pretty old version if your code doesn't work. Try the two options below. The first option is the standard way to calculate the double integral. If it doesn't work, then try the second option, which reduces the double integral to a repeated one:
 

restart;
r:=-36*u^3*v*(-1 + u + v)*(3*u^2 + u*v - 5*u - 2*v + 2)/(-1 + u)^2;
int(r, [u=0..1-v, v=0..1]); # The first option
int(int(r, u=0..1-v), v=0..1);  # The second option

Examples:

ilog10(10110)+1;
ilog10(111010110)+1;

                                                        5
                                                        9

restart;
de := plot3d(y*sin(x), x = 0 .. 2*Pi, y = -2 .. 2):
plottools:-getdata(de);

restart;
ell:=(9/4)*X^2+9*Y^2=1;
a:=sqrt(1/coeff(lhs(ell),X^2));
b:=sqrt(1/coeff(lhs(ell),Y^2));
max(2*a,2*b); # Major axis
min(2*a,2*b); # Minor axis

restart;
z1:=2+3*I:
z2:=I:
op(z1);
op(z2);

                                                                         2, 3
                                                                          1

A reliable way to iterate over the sum of multiple terms is to simply make those terms elements of a list.

restart:
with(plottools): with(plots):
u:=[2,2]: v:=[2,-1]:
G1:=seq(line(-5*u+t*v,5*u+t*v,linestyle=2), t=-5..5):
G2:=seq(line(s*u-5*v,s*u+5*v,linestyle=2), s=-5..5):
U:=arrow(u, color=red, width=0.2):
V:=arrow(v, color=red, width=0.2):
plots:-display(U,V,G1,G2, size=[700,700], scaling=constrained, axes=none);
    

We have to use the  parametric  option:

restart;
solve({-x1+2*x2+(2-p)*x3=0,(2-p)*x2+x3=2, (1-p)*x1+2*x2+2*x3=p+3}, {x1,x2,x3}, parametric=true):
value(%);

Your syntax is correct and works successfully, but the following small improvements can be made:

1. The inequality  y>-1  can be omitted, since your region is itself above the line  y=-1 .
2. The  optionsexcluded = (color = white)  option can also be omitted, as it is done by default.
3. I equalized the ranges along the axes and shortened them a bit. With equal ranges, all shapes are unchanged and the  scaling=constrained  option is not required.

restart: with(plots):
inequal({y >= x^2+1, (x-1)^2+(y-1)^2 <= 16}, x = -3.5 .. 5.5, y = -3.5 .. 5.5, optionsfeasible = (color = grey));

The  solve  command is not intended for solving matrix equations. You can solve your problem using a simple for-loop. Note that to check for matrix equality, we must use the  LinearAlgebra:-Equal command:

Download Mapleprimes_Question_Book_2_Paragraph_4.1_Question_9_new.mw

Use the  factor  command for this:

factor(f);

                    A*sin(x)*theta(x)*(-m*omega^2+k)

By default, Maple calculates with 10 significant digits. Therefore, for a sufficiently large  x  of float type, we get ln(x+1) - ln(x) = 0. . If we increase the accuracy of the calculations, then the error disappears:

restart;
f := x->4*x*(ln(x+1)-ln(x)):
Digits := 50: 
evalf[10]~([seq(f(10.^n), n = 1 .. 10)]);

  [3.812407192, 3.980132341, 3.998001332, 3.999800013, 3.999980000, 3.999998000, 3.999999800, 3.999999980, 3.999999998, 4.000000000]

We can simply calculate this determinant and, equating it to  0 , we get a 4th degree equation for  w  with 2 parameters  d  and  k . Giving some values to these parameters, we find all values of  w . In the example below we find the real  d=2, k=1, w=sqrt(3)/3  for which  the determinant  is  0 :

Download det_new.mw

restart;
with(Student[Calculus1]):
soln := Roots(y-0.4646295e-3*tanh(y)+0.1839145082e-2*tanh(y)/(0.6000000000e-3*y^2-0.1840000000e-2), y);
map(t->t*I, soln);