@want to be a permanent vegan  The only new issue in your new question - is finding the integral of the function, which occurs in the numerical solution of the system of differential equations (also the initial conditions must be present). To find the integral add to your system an additional function, using the fact that  int(f(t), t=x0..x) = g(x)  is equivalent to 2 equalities  diff(g(x),x) = f(x)  and  g(x0) = 0 

@tomleslie  The same can write shorter:

f:=x->x*cos(x)-sin(x)*sin(x/1000);
r:= 0:
to 20 do
r:=RootFinding[NextZero](f, r);
od;

@snowww  

restart;

N := {F = 1, K = 1.2}:

Eq := diff(C(x),x,x)-F*(diff(C(x),x))-F*K*C(x) = 0:

Bcs := F*D(C)(0) = C(0)-1, D(C)(1) = 0:

Sol := dsolve(eval({Bcs, Eq}, N), C(x)):

assign(%);

plot([C(x), eval(exp(-K*x), N[2])], x = 0 .. 3, 0 .. 3, color = [red, blue], thickness = 2);

@snowww  You have to love Maple and work with it constantly.

@farzane  The above code works well in this case also:

A:= .5464691235-.4473247264*I,  -.4563184747+1.*10^(-14)*I,  .5464691235+.4473247264*I, 0.9091925189:

Re(sort([A], (z1,z2)->abs(Im(z1))<abs(Im(z2)))[1]);

                                    0.9091925189

@Markiyan Hirnyk  You're right of course. Think of it as a joke!

@Markiyan Hirnyk 

P:=a*b^2+sqrt(41)*b*c-a^3*c^2;

P1:=subs(sqrt(41)=d, a*b^2+sqrt(41)*b*c-a^3*c^2);

Q:=expand(P1^7);

ind:=indets(%) minus {d};

n:=igcd(seq(degree(Q,s), s=ind));

subs(d=sqrt(41), RealDomain[simplify](Q^(1/n), symbolic));

@Markiyan Hirnyk 

1) I'm not familiar enough with the theory of polynomial ideals and radicals in them. As I understand it from the original post, it is necessary for a given polynomial in several variables over real numbers just to find the root from it of maximum order, of course, if this root there exists.

2)  I did not compare my method with other methods of solving this problem.

3) I just answered that question OP   "Is there another commands for this?"

@MDD  This means that an assumption is imposed on the variable a. If you want the symbol tilda to be absent, click

Tools->Options->Display->Assumed variables->No Annotation-> Apply Globally

@MDD  You must (in the copiable form) provide the full text of the code in which this error occurs.

@MDD   it is.

@Markiyan Hirnyk 

A := [x, y, x^2*y, x*y^2, y^2]:

B := [x^2, y^3]:

remove(c->`or`(seq(divide(c, b), b = B)) , A);

                                [x, y, x*y^2, y^2]

@Carl Love  Thank you very much for your response. This is a very effective way!

@Markiyan Hirnyk   for your interest! I meant that a segment is a closed interval. Of course easily to adjust procedure for all cases.

@Carl Love  I do not understand why on your images  additional lines appears, for example for k=1 .  Compare with my image:

F:=(x,y)->(1/2*x^2-3/2*y^3,-1/2*x^3+1/3*y^2):  # Nonlinear mapping R^2->R^2

X1:=t->(t,0): X2:=t->(0,t): X3:=t->(t,1): X4:=t->(1,t):

plot([[X1(t),t=0..1], [X2(t),t=0..1], [X3(t),t=0..1], [X4(t),t=0..1], [F(X1(t)), t=0..1], [F(X2(t)), t=0..1], [F(X3(t)), t=0..1], [F(X4(t)), t=0..1]], color=[red$4,blue$4], thickness=3, scaling=constrained);