Actually you can also use Heaviside via piecewise like this:

a0(t):=piecewise(x(t)+a[1](t)>0,-x(t),a[1](t));
a0(t):=convert(a0(t),Heaviside);
This has the same effect of making it possible for convertsys to do its job of isolating the highest derivatives.

Actually you can also use Heaviside via piecewise like this:

a0(t):=piecewise(x(t)+a[1](t)>0,-x(t),a[1](t));
a0(t):=convert(a0(t),Heaviside);
This has the same effect of making it possible for convertsys to do its job of isolating the highest derivatives.

Since you found no problem in Maple with the parameter values mentioned maybe you could give us an example of a pair of parameter values zeta and n for which problems do exist in Maple.

@acdah The way you proceed it does not make sense to assign values to theta, r, epsilon.
That was  really meant by saying you were stuck.

If you want approximations they should take place before integration. So approximate the integrand.

@acdah  Being of type numeric means being actual numbers, i.e. integers, rationals, or floating point numbers. So you cannot change the types of the variables, but you can assign values of type numeric to them, as in alpha:=1.2345;

You have an unevaluated integral in mH from the integral in mean_integral, where you have something like
Int( f(r,alpha,beta,epsilon,theta,M), theta=0..Pi,method= _Gquad).
_Gquad is a numerical method and would require r,alpha,beta,epsilon,M all to be of type numeric, but in your expression they are names.
If you replace the definition by
mean_integral := 2*Pi*int(simplify(mean_square)*area_element, theta = 0 .. Pi);
you will see that it returns unevaluated. I think you are stuck there.

@N00bstyle Your procedure reductiefactorvoetgangershoogte is unnecessarily complicated and involves a lot of calls to eigenfrequentieigvvoetgangers. But in this case the main problem is that evalb (which is used in the if statements) cannot handle Pi:
try
if Pi<3 then 7 else 0 end if;
evalb(Pi<3);
You could use 'is' instead of evalb as in
if is(Pi<3) then 7 else 0 end if;
is(Pi<3);
Instead you could replace Pi by evalf(Pi).
So a suggestion for your procedure reductiefactorvoetgangershoogte would be to change the procedure to
reductiefactorvoetgangershoogte := proc (_hoogte)
local eg; eg := evalf(eigenfrequentieigvvoetgangers(_hoogte));
if eg <= 1.25 then 0 elif 1.25 < eg and eg < 1.7 then (eg-1.25)/(1.7-1.25) elif 1.7 <= eg and eg <= 2.1 then 1 elif 2.1 < eg and eg < 2.3 then (2.3-eg)/(2.3-2.1) elif 2.3 <= eg and eg <= 2.5 then 0 elif 2.5 < eg and eg < 3.4 then .25*(eg-2.5)/(3.4-2.5) elif 3.4 <= eg and eg <= 4.2 then .25 elif 4.2 < seg and seg < 4.6 then .25*(4.6-eg)/(4.6-4.2) elif 4.6 <= eg then 0 end if end proc;
(I'll let you do the indenting)
My points are: eigenfrequentieigvvoetgangers(_hoogte) is evaluated only once and then evalf is aplied.
However, I still think you should use unapply and piecewise instead of your present approach.
Why insist on making it complicated?

@N00bstyle Your procedure reductiefactorvoetgangershoogte is unnecessarily complicated and involves a lot of calls to eigenfrequentieigvvoetgangers. But in this case the main problem is that evalb (which is used in the if statements) cannot handle Pi:
try
if Pi<3 then 7 else 0 end if;
evalb(Pi<3);
You could use 'is' instead of evalb as in
if is(Pi<3) then 7 else 0 end if;
is(Pi<3);
Instead you could replace Pi by evalf(Pi).
So a suggestion for your procedure reductiefactorvoetgangershoogte would be to change the procedure to
reductiefactorvoetgangershoogte := proc (_hoogte)
local eg; eg := evalf(eigenfrequentieigvvoetgangers(_hoogte));
if eg <= 1.25 then 0 elif 1.25 < eg and eg < 1.7 then (eg-1.25)/(1.7-1.25) elif 1.7 <= eg and eg <= 2.1 then 1 elif 2.1 < eg and eg < 2.3 then (2.3-eg)/(2.3-2.1) elif 2.3 <= eg and eg <= 2.5 then 0 elif 2.5 < eg and eg < 3.4 then .25*(eg-2.5)/(3.4-2.5) elif 3.4 <= eg and eg <= 4.2 then .25 elif 4.2 < seg and seg < 4.6 then .25*(4.6-eg)/(4.6-4.2) elif 4.6 <= eg then 0 end if end proc;
(I'll let you do the indenting)
My points are: eigenfrequentieigvvoetgangers(_hoogte) is evaluated only once and then evalf is aplied.
However, I still think you should use unapply and piecewise instead of your present approach.
Why insist on making it complicated?

@N00bstyle If after having defined eigenfrequentie in your version, you try the command
eigenfrequentie;
then you reveive the error
Error, (in qigvvoetgangers) cannot determine if this expression is true or false: 0 <= 0.153846153846154e-1*(65.0*hoogte+.967200000000000)/hoogte-21/20
That is where you have to ask yourself what to do. In fact you must have noticed the problem yourself (I guess) since you don't use eigenfrequentie in the plot command defining ploteigenfrequentie, but rather the expression used to define eigenfrequentie.

@N00bstyle If after having defined eigenfrequentie in your version, you try the command
eigenfrequentie;
then you reveive the error
Error, (in qigvvoetgangers) cannot determine if this expression is true or false: 0 <= 0.153846153846154e-1*(65.0*hoogte+.967200000000000)/hoogte-21/20
That is where you have to ask yourself what to do. In fact you must have noticed the problem yourself (I guess) since you don't use eigenfrequentie in the plot command defining ploteigenfrequentie, but rather the expression used to define eigenfrequentie.

@Markiyan Hirnyk Try

inttrans[invlaplace](s^(-2.3)/(s+c),s,t); #No problem
inttrans[invlaplace](s^(-2.3)*cos(s)/(s+c),s,t); #Not found
That the last one is not found is just because invlaplace isn't perfect.

In the question p was assumed constant. However, it may be worth mentioning that the non-constant case can be handled as well, but we must use DEtools[mult]:

with(DEtools):
L:=(Dx-p(x))^2;
diffop2de(L,f(x),[Dx,x]); # does not correspond to composition
#This gives the correct version:
mult(Dx-p(x),Dx-p(x),[Dx,x]);
diffop2de(%,f(x),[Dx,x]);

In the question p was assumed constant. However, it may be worth mentioning that the non-constant case can be handled as well, but we must use DEtools[mult]:

with(DEtools):
L:=(Dx-p(x))^2;
diffop2de(L,f(x),[Dx,x]); # does not correspond to composition
#This gives the correct version:
mult(Dx-p(x),Dx-p(x),[Dx,x]);
diffop2de(%,f(x),[Dx,x]);

@GOODLUCK If you are rewriting your system using other names, then be sure to avoid gamma, Pi, I, and D, since they have special meanings in Maple:
gamma is Euler's constant, try evalf(gamma);
As said earlier Pi is the usual, try evalf(Pi);
I stands for sqrt(-1), try I^2;
D is the differentiation operator, D(sin); and D(sin)(Pi/3); #compare to diff(sin(x),x);
Also avoid using a name like alpha together with indexed versions of alpha like alpha[0]. Say you assign to alpha[0], alpha[0]:=7.913. Then you have implicitly created a table with name alpha, try eval(alpha);
Also use * as the multiplication sign, the dot is really meant for matrix multiplication.

@GOODLUCK Notice the tilde in my command
fvars:=freeze~(vars);
That means that the procedure freeze will be applied to each element in the set (or list or other "container") vars.
This syntax for elementwise operation was introduced in Maple 13.
See
?elementwise
If instead you just do freeze(vars); then you freeze the whole set of variables.
Notice the different outputs of
freeze(vars); #returns one "freeze" name
and
freeze~(vars); #returns a set of "freeze" names and also regular names (these are not frozen as there is no need for that).