You can use a common  if ... fi:  for this:

restart;
str:="A";
x:=10;
if x=10 then str:=cat(str," it was 10"); x:=11 else
str:=cat(str," it was not 10"); x:=9 fi;

Edit.

See help on the  evala  command. Also the  Algebraic  package may be useful for you.

You forgot to call the package  plots . Insert  with(plots):  at the beginning of your worksheet.

Replace in the code the dot  .  with the multiplication sign  * .

The corrected file 

test1_03_new.mw

The error you receive says that Maple does not have the  FourierSeries  package.

You can freely download the  OrthogonalExpansions  package from the Maple Application Center. This package contains a number of commands for working with Fourier series.

https://www.maplesoft.com/applications/view.aspx?SID=33406

(rhs@op)~([rootsq0]);

There will be infinitely many positive solutions too. Below we find the smallest:

restart;
eq := 61*x^2 + 1 = y^2;
[isolve(eq)];
map(t->eval([x,y],t),%);
map(t->expand(eval(t,_Z1=1)), %);
[x,y]=~select(t->t[1]>0 and t[2]>0, %)[];

                     [x = 226153980, y = 1766319049]

Edit. It is convenient to find the desired solutions using the procedure:

restart;
Eq := 61*x^2 + 1 = y^2:
Sol:=n->subsindets(eval(isolve(Eq)[4],_Z1=n),realcons,expand):

# Examples of use:
Sol(1);
Sol(2);

In fact, your example is on optimization with a parameter  a . The commands of simbolic optimization  minimize  and  maximize  can solve such examples only in very simple cases, for example  minimize(a*x+3, x=0..1);  , a little more difficult   minimize(x^2+a*x+3, x=0..1);   - it already fails. Below is a solution using a procedure  Min  that returns the result for each specific  a :

restart;
local f;
Min:=proc(a)
uses Optimization;
f:=a*x + log10(3+4.2^x +2.2^(2*x));
if a>=0 then return [evalf(eval(f,x=0)),[x=0]] else
Minimize(f, x=0..1) fi;
end proc:

# Examples:
Min(2);
Min(-2);
Min(-0.3);

You did not specify the base of the logarithm, I took it equal to  10 .

This is not an integral equation, but simply a numerical calculation of a definite integral with the parameter  t :

restart; 
rho := 1/4: mu := 1/4: T := (t-x)^rho: 
E := add(T^k/GAMMA(k*rho+mu), k = 0 .. 5);
h :=unapply(int(E*(8*x^(3/2)/(3*sqrt(Pi))+x^2+(1/2)*exp(-x)-8/3)/(t-x)^.75, x = 0 .. 1, numeric), t):

# Examples:
h(1.5);
plot(h, 1..2, labels=["t","h(t)"]);

Another way is shown below with a simple example:

restart;
Sol:=dsolve({diff(y(x),x)=x+y(x), y(0)=0}, numeric);

f:=s->eval(y(x),Sol(s));
f(1);
int('f(s)',s=0..1, numeric);

f := g->int(g(r+eps), r);

f(h);

There are no such numbers, because your equality  sqrt(a+b*sqrt(c+d*sqrt(e +f*sqrt(g)))) = h  implies that  g  is an exact square:

g=solve(sqrt(a+b*sqrt(c+d*sqrt(e +f*sqrt(g)))) = h, g);

The graph shows that the root is close to 0 :

restart;
Digits:=20:
y:=t->808.2213240*(1 - 0.63*(1993551437/1601983488 - sqrt(3)/2)^0.3)*(1 - 335345*(45188/147189 - 53/(4820*ln(2)))*335345^(131537/203808)*131537^(72271/203808)*(1 - 1/(1 + (203808*exp(-677.0138344*t))/131537)^(131537/203808))/34603964738):
plot(y, -0.1..0.1);

fsolve(y(t)=196.9594856, t=-1..1);

restart;
A:=fsolve(2*x^5-x^4-0.5*x^3+3*x^2-0.5);           
B:=fsolve(x^2-1);
sort([A,B])[];

                                  -1.,  -1.000000000,  -0.4158624444,  0.4256094086,  1.

To create an animation, it is convenient to first create a procedure that creates one frame (in the future, the procedure parameter will be an animation parameter). Below are 2 procedures. The  Cone  procedure creates the cone with the base radius  r , height  h  and base center  c . The second procedure  Plane  creates the plane  z=h-y  intersecting the axis  Oz  at a point  [0,0,h] :

restart;
Cone:=proc(r,h,c:=[0,0,0])
uses plottools, plots;
display(cone(c,r,-h, color="LightBlue"), scaling=constrained, axes=none);
end proc:
Plane:=(h,X,Y)->plot3d(h-y,x=-X..X,y=-Y..Y, color=yellow):

# Examples of use

plots:-display(<Cone(2,2)|Cone(2,3)|Cone(2,5)>);
plots:-animate(Cone,[2,h,[0,0,h]], h=1..5, frames=60);
plots:-animate(Plane,[h,1.8,1.8], h=5.7..2.5, background=Cone(2,6,[0,0,6]), frames=60);