Let's help Maple a little. If  t^2 - 4 = s^2 - 4  and  t<>s  then  s = -t . Below we get that there will be infinitely many double points:

restart;
x:=t->sin(t): y:=t->t^2-4:
solve({sin(t)=sin(s),s=-t}, AllSolutions);
eval(%, _Z1=k);
[x,y]=eval([x(t),y(t)], %) assuming k::posint;  # All the double points
# k = 1, 2, 3, ...

                               [x, y] = [0, Pi^2*k^2-4]

The numerical solution:

restart;
x:=t->sin(t): y:=t->t^2-4:
Sys:=[x(t)=x(s),y(t)=y(s)]:
plots:-implicitplot(Sys, t=-10..10,s=-10..10, color=[red,blue], gridrefine=3);
fsolve(Sys, {t=-4..-1,s=1..4});
fnormal(eval([x(t),y(t)], %));

                                    [0., 5.869604404]

Because there are many solutions, then the plot is needed to isolate the root.

With local, you made a new name (it's not clear why?) from the command  pdetest . Just replace the line local pdetest := (ans, de1)  with the line  test := pdetest(ans, de1)

restart;
did:=D^14+23*D^13+144*D^12-30*D^11;
subsindets(did,symbol^integer,t->(D@@degree(t))(y)(x));
convert(%, diff);

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);