I used an alternative method that makes use of Matlab's fzero function. You can also see one way of avoiding the problem at x = 0:
xlow=0; xhigh=0.5;
xspan = [xlow xhigh];
dydx0 = 0;
y0 = 0; % Initial guess for y(x=0)
% Use fzero to get value of y0 that makes y(x=0.5) = 0.1;
y0 = fzero(@(y0) y0val(y0, xspan), y0);
disp(['y(x=0) = ' num2str(y0)])
% Now use ode45 to integrate from x=0 to x=0.5 with good starting value
% for y(x=0)
Y0 = [y0 0];
[x, Y] = ode45(@odefn, xspan, Y0);
plot(x,Y(:,1)),grid
xlabel('x'),ylabel('y')
function Z = y0val(y0, xspan)
Y0 = [y0, 0];
[~, Y] = ode45(@odefn, xspan, Y0);
yend = Y(end,1);
Z = yend - 0.1;
end
function dYdx = odefn(x, Y)
y = Y(1); dydx = Y(2);
d2ydx2 = 0.64*y;
if dydx>0
d2ydx2 = d2ydx2 - (2/x)*dydx;
end
dYdx = [dydx;
d2ydx2];
end
