@Carl Love Thanks. Sometimes I hope the OPs provide at least some information to help. (If it's not in Horn and Johnson it seems it must be a bit exotic.) Turns out it was easier than I thought to calculate.

Please explain what a pseudospectrum is. If you just mean to plot the eigenvalues of a matrix in the complex plane, that can easily be done with complexplot.

@SHIVAS You need to solve and create the plots in a loop and then combine them with plots:-display. For example if your plots for different Nr are eta[1],eta[2] etc, then they are combined by something like plots:-display([seq(eta[i],i=1..3)])

@treverona Once you have a symbolic power like your d then Maple doesn't consider it as a polynomial, so the usual commands to find degrees don't work.

@sursumCorda OP said multivariable polynomial, and I think it is OK for that. I suppose it would be more robust if you expanded p first, but the meaning of the answer depends on the order of the terms that is given, so I assumed that it was given as a sum of terms.

@SHIVAS Ans[k] is the output from pdsolve, which is a module - it can't be used in the same way as the output from dsolve. See the help page ?pdsolve,numeric for how to use the module to create plots and find values.

@Tamour_Zubair 

solve(subs(y(t)=y,sol),t); gives t as a function of y.

@Tamour_Zubair There isn't an analytical solution for y as a function of t; the RootOf is the best you can do. On the other hand, from the implicit solution you can get t as an analytical function of y. 

Please check your boundary condition D(f)(0) = lambda + xi*(D^2)(f)(0). Note that if you wanted the second derivative and not the square of the first derivative, then you need D(f)(0) = lambda + xi*(D@@2)(f)(0)

@NeraSnow Your procedure to upload the worksheet was correct, but the website is not working correctly today, as happens sometimes.

@Carl Love Yes, I tried something like this expecting the new inline loops might be more efficient, but was disappointed (as I've found a couple of times before).

There is also the issue of counts>9, which I missed originally. So "1111111111111" gives "131" in my code, but "11111111111111" in yours.

@sursumCorda Arrays are usually efficient, but here adding on to the end with the ,= operation is part of the secret.

@AngieS7 It means it can't find a solution. You can help it by providing ranges (or an initial guess), but it might mean there is no solution

@AngieS7 You can tell fsolve to only look for solutions in specific ranges by using something like

fsolve({eq1, eq2, eq3, eq4, eq5, eq6}, {A=0..5, B=3..7, C=0..infinity, omega=0..10, x1=0..1, x2=0..1});

@AngieS7 It is hard to tell why this hasn't worked without you uploading your worksheet. You can do that with the green up-arrow in the Mapleprimes editor.

Edit: looks like you might have changed vb:=0.5 to vb:=5