Why do I receive "Array indices must be positive integers or logical values"?

clc; clear all; close all; 
syms epsr_bar eps_r  r
r = 0.001;
a = 1;  f = 0.05;
n = 0.1;
eps2 = 1;
A = linspace(0,4,20);
for x = 1:length(A)
    eps_r = A(x).*(r/a).^n;
    cond = epsr_bar(0.001) == eps_r(0.001);
    eqn = diff(epsr_bar(r),r) == (2.*eps_r.^2-eps_r.*epsr_bar-epsr_bar.^2)./(r.*eps_r);
    ySol(r) = dsolve(eqn,cond);
    eps_e(x) = eps2+((3*eps2*f).*((ySol-eps2)./(ySol+2*eps2)));
end
plot(A,eps_e)

Answer 1

[Image Analyst]

It's thoroughly explained in the FAQ:

https://matlab.fandom.com/wiki/FAQ#.22Subscript_indices_must_either_be_real_positive_integers_or_logicals..22

Answer 2

[Walter Roberson]

syms epsr_bar eps_r  r

That makes epsr_bar into a symbolic scalar variable, not a function. It also creates eps_r and r as scalar symbolic variables.

r = 0.001;

That overwrites r with a numeric (non-symbolic) value.

cond = epsr_bar(0.001) == eps_r(0.001);

epsr_bar is a symbolic variable, not a symbolic function, so epsr_bar(0.001) is an attempt to index the symbolic variable at location 0.001, which is an invalid location.

[Walter Roberson]

syms epsr_bar(r)
r0 = 0.001;
a = 1;  f = 0.05;
n = 0.1;
eps2 = 1;
A = linspace(0,4,20);
num_a = length(A);
eps_e = cell(num_a, 1);
for x = 1:num_a
    eps_r(r) = A(x).*(r/a).^n;
    cond = epsr_bar(r0) == eps_r(r0);
    eqn = diff(epsr_bar(r),r) == (2.*eps_r(r).^2-eps_r(r).*epsr_bar(r)-epsr_bar(r).^2)./(r.*eps_r(r));
    ySol(r) = dsolve(eqn,cond);
    eps_e{x} = eps2+((3*eps2*f).*((ySol(r)-eps2)./(ySol(r)+2*eps2)));
end

Warning: Unable to find symbolic solution.

eps_e

eps_e = 20×1 cell array
    {0×0 sym                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         }
    {[(3*((4*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) - 1))/(20*((4*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) + 2)) + 1  ]}
    {[(3*((8*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) - 1))/(20*((8*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) + 2)) + 1  ]}
    {[(3*((12*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) - 1))/(20*((12*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) + 2)) + 1]}
    {[(3*((16*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) - 1))/(20*((16*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) + 2)) + 1]}
    {[(3*((20*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) - 1))/(20*((20*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) + 2)) + 1]}
    {[(3*((24*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) - 1))/(20*((24*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) + 2)) + 1]}
    {[(3*((28*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) - 1))/(20*((28*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) + 2)) + 1]}
    {[(3*((32*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) - 1))/(20*((32*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) + 2)) + 1]}
    {[(3*((36*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) - 1))/(20*((36*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) + 2)) + 1]}
    {[(3*((40*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) - 1))/(20*((40*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) + 2)) + 1]}
    {[(3*((44*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) - 1))/(20*((44*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) + 2)) + 1]}
    {[(3*((48*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) - 1))/(20*((48*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) + 2)) + 1]}
    {[(3*((52*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) - 1))/(20*((52*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) + 2)) + 1]}
    {[(3*((56*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) - 1))/(20*((56*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) + 2)) + 1]}
    {[(3*((60*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) - 1))/(20*((60*r^(1/10)*(921*(1/1000)^(921^(1/2)/10) + 29*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 29*(1/1000)^(921^(1/2)/10)*921^(1/2)))/(19*(921*(1/1000)^(921^(1/2)/10) + 31*921^(1/2)*r^(921^(1/2)/10) + 921*r^(921^(1/2)/10) - 31*(1/1000)^(921^(1/2)/10)*921^(1/2))) + 2)) + 1]}

But now what? You have constructed an array of functions of r, one for each value of A, but to plot them you need to know the range of r to plot over. You do not want to plot only at the fixed point r0 (0.001), as your cond forces the value at r0 to be something particular, and there would be no point in solving the differential equation if you only wanted to plot the fixed value.