Need to set the right condition. For exp(t), the condition is s-1>0. For general exp(a*t) the condition is s-a>0. 

Download lap.mw

in addition to what was said in other replies.

The issues seems that Maple could not integrate 

Using another software, this has antiderivative

Copying this result to Maple to give final solution and odetest gives zero. Here is the worksheet

Download dsolve_9_5_2020.mw

subexp := sin(omega*t + alpha)*sin(omega*t + phi);
subexp2:= M__a*I__a*trigsubs(subexp)[1];

You can't solve the above since there is no equation. An equation must have a `=` in it and your's do not. May be this is why it did not work when you tried it?

If you can post an actuall differential equation, then I am sure Maple will be able to solve it, even if the diff was in the denominator.

For example

restart;
dsolve(y(t)/diff(y(t),t)=1)

gives

                        y(t) = _C1*exp(t)

 in the example I would expect it to return exp(x).

Did you try to enter it in Maple?

sum(x^n/n!,n=0..infinity)

And Maple returns exp(x)

why can't you just write

a:=3;
plot(x, x = 0 .. a)

but if you do not want, you can try

f:=a->plot(x, x = 0 .. a);
f(3)

But if you do not want, you can try

restart;
eval('plot(x, x = 0 .. a)',a=3)

But if you do not want, you can try

restart;
simplify('plot(x, x = 0 .. a)',{a=3})

OP said

I found another solution I like:

use a=3 in plot(x, x=0..a) end use;

If you want to do the above, then you could also do

..... 

proc() 
   local a=3; 
   plot(x,x=0..a); 
end proc 
    ()
.....

The above introduces local block inside you other functions and a inside is only local to that block. Same as with use but I think is better.

I learned the above from Carl answer here https://www.mapleprimes.com/questions/201123-Can-One-Declare-Local-Variables-With 

there is none. Even this, without the sqrt, has no anti-derivative

restart;
p :=1/(ln(t)^2 + t);
int(p,t)

restart;
f := Statistics:-RandomVariable(Normal(1,1/2)):
plot(Statistics:-PDF(f,t),t=-3..3);

ode:=m*diff(x(t),t$2)+c*diff(x(t),t)+k*x(t)=F(t);
ic:=x(0)=1,D(x)(0)=0;
m:=1;c:=1/100;k:=10;
F:=t->Statistics:-PDF(f,t);
sol:=dsolve([ode,ic],x(t));

plot(rhs(sol),t=0..20)

where the expression would be M = 2Krst / w

one way could be

restart;
M:= 2*p*q*r*s*t /(u*v*w);
algsubs(p*q/(u*v)=k,M)

Another way

restart;
M:= 2*p*q*r*s*t /(u*v*w);
solve(p*q/(u*v)=k,p);
subs(p=%,M)

you really do not need Maple for this

Check using Maple:

ode:=diff(f(x),x)=f(x)^(-1);
dsolve(ode);

You mean code block?  You can do the following

This will open a window

Now type the code inside it, then click OK

That is all.  To edit the code again, repeate the above process.

Here is one way (need Maple 2020 for the Slice command)

A:=Matrix(3, 5, [[2, -6, 3, 0, 0], [5, -2, 4, 1, 2], [17, -4, 10, 20, 99]]);
ListTools:-Slice(convert(A^+,list),LinearAlgebra:-RowDimension(A))

Here is another way

convert~([seq(A[i],i=1..LinearAlgebra:-RowDimension(A))],list);

split_expr:=proc(expr::anything)
 if type(expr,`+`) then
    return convert(expr,list);
 elif type(expr,`list`) then
    return expr;
 else
    return [expr];
 fi;
end proc;

create file data.csv  (ps. you have 6 variables actually not 5, but you can change this)

2,-6,3,0,0
5,-2,4,1,2
17,-4,10,20,99

Then write

restart;
currentdir("C:/tmp"); #where the data.csv file is
data := Import("data.csv"):
data := convert(data,Matrix):
nRows,nCols := LinearAlgebra:-Dimension(data);
for n from 1 to nRows do
    unassign('a,h,b,f,g'):
    assign~([a,h,b,f,g],convert(data[n],list)):
    Conic:=a*x^2+2*h*x*y+b*y^2+2*f*x+2*g*y+c;
od;

I can't belive it! No Fourier Series in Maple!!! What a shame!

sorry, I didn't realize OP wanted solution using only Maple build-in commands and not using external package OrthogonalExpansions. Will try to find if I can find build-in Maple command to do this if possible.