How to identify stiffness of a second order non-linear ode How to identify stiffness of this equation
$$\frac{1}{2}F''+\frac{x}{2}F'+F-\frac{1}{2}F^2=0$$
$F'(0)=0$ and $1<F(0)=Constant<2$ 
This problem comes from trying to obtain a numerical solution to this equation. In choosing ode solver in MATLAB I came across this concept of stiffness. ode113 is described for non-stiff equations, and the MATLAB gives a seemingly good result. But I don't know if it is right to use this. Because if it's stiff then I should probably use ode23d as documented
 A: Stiffness concerns how 'hard' a problem is to solve numerically. If a problem is stiff, it typically means that you would have to use a very small time-step in an explicit scheme to solve it without seeing spurious instabilities. This means that you will be waiting for a while to simulate out to a reasonable time. A stiff solver is more stable somehow (typically by being implicit/semi-implicit) and allows you to take a larger time-step. MATLAB does all of this time-step selecting business for you 'under the hood' unless you provide it with input options. The only thing it cant really do for you is select the most optimal scheme to solve your problem. That's what choosing ODE115s, or ODE45, or ODE23d, is all about. If you are getting a solution from any of these methods, regardless of the stiffness of your problem, it is correct (up to different numerical accuracies for different schemes), you just may have waited longer than you needed to.    
A: If you consider a fixed-step solver, then the error of the solution has essentially 3 modes. For large step sizes the error is non-linear and chaotic due to non-linearities of the equation. Then for medium small step sizes the error behaves approximately as the order dictates, that is, it is of size $C·h^p$ with an almost constant $C$. That is, to this point if you graph the error divided by $h^p$ you get a function that seemingly converges to a constant.
However, for very small step sizes, the accumulation of floating point errors over the single step dominates, the error is random with a mean of size $O(1/h)$. 
An initial value problem can be considered "stiff" if it does not have that middle "working" interval of step sizes, that is, you move directly from the randomness due to the non-linear nature of the problem to the randomness of floating point error accumulation.
The step size where floating point error accumulation becomes dominant depends on the precision of the floating point format, problems that are stiff with the above description in 32bit single precision might be workable in 64bit double precision.
Note that this is not an exact definition, more a description of the idea of a stiff IVP. As far as I know, there are no strict definitions of this concept.
