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Before entering the for-loop, define;

mycolors := ["black", "blue", "red", "pink"];

Then specify the color in the for-loop:

for k to 4 do
  R := dsolve(eval({bc, sys}, Ha = L[k]), fcns, type = numeric, AP);
  AP := approxsoln = R;
  p1u[k] := odeplot(R, [y, U(y)], 0 .. 1, numpoints = 100,
      labels = ["y", "U"], style = line, color = mycolors[k]);
end do;

Alternatively, you may omit the color option in the loop, and apply the colors to the composite plot, as in;

display([p1u[1], p1u[2], p1u[3], p1u[4]], color=mycolors);

but note that in the first argument of display I have changed your curly braces to square brackets.

Kitonum has already shown how to compute the derivative for a specific function f. I just want to add that specifying the f is not necessary.  It can be done for any f:

restart;
D[1](f)(z*sqrt(a/nu[f]), U*t/(2*x));

restart;
with(Student[MultivariateCalculus]):
r := (phi,theta) -> <(cos(phi)+3)*cos(theta), (cos(phi)+3)*sin(theta), sin(phi)>;
SurfaceArea(Surface(r(phi,theta), phi=0..2*Pi, theta=0..2*Pi));

The answer is 12 π².

I don't have GlobalOptimization, but that does not seem to be an impediment. The basic minimize() works quite well.

As to plotting, I don't understand the question.  Fun is a function of four variables.  How do you expect to plot a function of four variables?  Furthermore, how is that question related to the minimization issue?

After you have solved the PDE, as in sol := pdsolve(...), do
U := sol:-value():
U(2.0, 3.0);
U(2.0, 3.1);

restart;
M := 5;
F := Matrix(M+1,M+1):
for r from 2 to M+1 do
  for s from 1 to r-1 do
    if type(r+s, odd) then
      F[r,s] := 2^(k+1)*sqrt((2*r-1)*(2*s-1));
    end if
  end do
end do:
evalm(F);

That error message manifests a "misunderstanding" by Maple but it appears to be harmless.  Introduce a third parameter, k, and assign it an arbitrary value and Maple produces the correct result:

restart;
f:=x->piecewise(x>=-1/2 and x<=1/2,1);
eq:={diff(x(t),t)-sum(f(t-k*T),k=0..K)*x(t)=0,x(0)=1};
sol:=dsolve(eq, numeric, parameters=[T,K,k]);
sol(parameters=[1,3,-1]);  # the third argument is an arbitrary number
plots:-odeplot(sol, [t, x(t)], t=0..6,
    color=red, thickness=3, view=0..40);

Or with a more interesting choice of parameters:

sol(parameters=[2,2,-1]);  # the third argument is an arbitrary number
plots:-odeplot(sol, [t, x(t)], t=0..8,
    color=red, thickness=3, view=0..13);

Download mw.mw

Write the summation as

sum( irem(s+r,2) * whatever, s=1 .. r-1);

because irem(s+r,2) is zero when s+r is even, and one otherwise.

Asking for a phase portrait for your equation is not exactly meaningful.  The responses that you have received present graphs of solutions y(x) versus x, which is not the usual meaning of a phase portrait.  A phase portrait is a plot in the phase space.

The concept of the phase space, however, pertains to autonomous differential equations.  Your equation is non-autonomous, so the graphs that you have been presented with are remotely reminiscent of phase portraits but you should be aware that they are not phase portraits.

That said, here is what I would do if I were to plot a set of graphs of y(x) versus x (not a phase portrait).

>  restart; >  with(DEtools): >  de := diff(y(x),x)=x^2-y(x)^2; >  a := 2.0;  b := 3.0; >  ic := seq(y(s)=+a, s=-a..a, 0.3),       seq(y(s)=-a, s=-a..a, 0.3),       seq(y(-a)=s, s=-a..a, 0.3),       seq(y(+a)=s, s=-a..a, 0.3): >  DEplot(de, y(x), x=-b..b, [ic], y=-b..b,     linecolor=black, thickness=1, arrows=none);

Download mw.mw

Download mw.mw

Define φ and ψψ as you already have.

Then:

r := 2^k * M;
phi_vec := Vector[column](r, i-> phi[i]);
`&psi;&psi;_vec` := Vector[row](r, j-> `&psi;&psi;`[j]);
P := phi_vec . `&psi;&psi;_vec`;

Aside: The symbol `&psi;&psi;` in your calculations is too cumbersome and unnecessary. Why not use a single-letter symbol such as an uppercase Ψ insteaad?

Here is a worksheet that shows the general method of assembling a matrix from smaller matrices.

I must alert you, however, to your use of M^(−1) *~ K in your worksheet. That certainly IS NOT the same as the matrix product M^(−1) . K. Which of those constructions do you have in mind?

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