bvp4c claims singular Jacobian encountered, but I don't think so

Hi All, sorry if I'm missing something obvious. I'm solving a fairly simple second order ODE with boundary conditions:

D_FC = 0.00146; 
k_loss = 10^9; 
k_abs = 10^40; 

T = 1e-6; 

x0 = 0;
xf = T;

function dFEdx = odefunE(x,F)
%F = [f, f']
dFEdx=zeros(2,1);
dFEdx(1)=F(2); %df/dx=f'=F(2)
dFEdx(2) = (k_loss/D_FC)*F(1);
end

function residual = bcfunE(F0, FT)
%F0 = [f(0), f'(0)]
%FT = [f(T) f'(T)]
    residual = [(F0(2)-(k_abs/D_FC)*F0(1)) (FT(2)-diss_flux_T/D_FC)]; %for each boundary
end

initial_mesh = linspace(x0, xf, 100);
solinit = bvpinit(initial_mesh, [1 0]); %initial guess
solutionE = bvp4c(@odefunE, @bcfunE, solinit);

i.e. the ODE is f''(x) = a*f(x) for a constant a. If I solve it by hand I get a simple exponential, so I'm not sure why it's running into trouble.

I noted in previous posts the 'singular matrix' input option, but before introducing that I'd like to understand why this is a problem in the first place.

Thanks for any advice.