clear
format long
%%
% We eliminate the guesswork for $w(0)$ and let |shoot| do the work for us.
lambda = 0.6;
phi = @(r,w,dwdr) lambda./w.^2 - dwdr./r;   
a = eps;      % avoid r=0 in denominator
b = 1;

%%
% We specify the given and unknown endpoint values.
lval = [];  lder = 0;   % w(a)=?, w'(a)=0
rval = 1;   rder = [];  % w(b)=1, w'(b)=?

%%
[r, w, dwdx] = shoot(phi, [a,b], lval, lder, rval, rder, 0.8);
plot(r, w)
title('Correct solution')
xlabel('x')
ylabel('w(x)')
grid on

%%
% The value of w at r=1, meant to be exactly one, was computed to be
w(end)

%%
% The accuracy is consistent with the error tolerance used for |ode45| by
% |shoot|. The initial value w(0) that gave this solution is
w(1)