myzero = 0.00001;
q = 0.1;
mu = 1;
f[z_] = 1-z^2;
sol = NDSolve[{ f[z] (z y''[z] + y'[z]) == q^2 z y[z],
   y[1] == 0, y'[1] == mu}, y, {z, myzero, 1}]