Stiff differential equation where Runge-Kutta $4$th order method can be broken Is there a stiff differential equation that cannot be solved by the Runge-Kutta 4th order method, but which has an analytical solution for testing?
 A: How about "impractical"?  Here's an example from Numerical Recipes in C, Second Ed., Sec. 16.6:
$$u'=998 u+1998 v$$
$$v'=-999 u -1999 v$$
where $u(0) = 1$ and $v(0) = 0$.
Then 
$$u(x) = 2 e^{-x}- e^{-1000 x}$$
$$v(x) = -e^{-x} + e^{-1000 x}$$
Runge-Kutta 4th order would require a stepsize of less than $0.001$ for any accuracy, which is obviously aggravating for the one of the solutions which could be acquired with a far coarser step.
A: Cleve Moler, in this note, gives an innocuous-looking DE he attributes to Larry Shampine that models flame propagation. The differential equation is
$$y^\prime=y^2-y^3$$
with initial condition $y(0)=\frac1{h}$, and integrated over the interval $[0,2h]$. The exact solution of this differential equation is
$$y(t)=\frac1{1+W((h-1)\exp(h-t-1))}$$
where $W(t)$ is the Lambert function, which is the inverse of the function $t\exp\,t$. (Whether the Lambert function is considered an analytical solution might well be up for debate, but I'm in the camp that considers it a closed form.)
For instance, in Mathematica, the following code throws a warning about possible stiffness:
With[{h = 40}, y /.
     First @ NDSolve[{y'[t] == y[t]^2 - y[t]^3, y[0] == 1/h}, y,
                     {t, 0, 2 h}, Method -> {"ExplicitRungeKutta",
                                             "DifferenceOrder" -> 4}]]

