Determine the minimum upper bound of $h$ for the classical 4th-order Runge-Kutta method to be absolutely stable for this problem. 
Question: Consider the initial value problem (IVP),
$$f(x,y) = y(23.51 - y), \quad y(0) = 12.$$
The exact solution of the problem increases from $y(0) = 12$ to $y = 23.51$ as $x$ increases without limit.


Determine the minimum upper bound of $h$ for the classical 4th-order Runge-Kutta method to be absolutely stable for this problem. Give your answer to 3 decimal places.

While solving the IVP, I obtain the exact solution is
$$23.51 \ln|y| - y = x + 46.4202.$$
However, I do not see that as $x$ tends to infinity, $y$ tends to $23.51$.
I also have trouble solving the second part of the problem.
Any hint is appreciated.
 A: Your solution looks wrong, you could treat it for instance as Bernoulli equation so that
$$
-(y^{-1})'=23.51y^{-1}-1\implies 23.51y(t)^{-1}-1=(23.51y(0)^{-1}-1)e^{-23.51t}
$$
from where you can easily extract the solution.
For the stability note that close to $y=23.51$ the equation is similar to
$$
y'=23.51(23.51-y)
$$
with "eigenvalue" $\lambda=-23.51$ and that the stability region of RK4 is given by $z= λh$ where
$$
|1+z+\tfrac12z^2+\tfrac16z^3+\tfrac1{24}z^4|\le 1.
$$
The boundary you look for is thus where $z=-23.51h$ is a real negative root of
$$
2+z+\tfrac12z^2+\tfrac16z^3+\tfrac1{24}z^4=0
$$
which only has complex roots or
$$
z+\tfrac12z^2+\tfrac16z^3+\tfrac1{24}z^4=0
$$
which has $-2.78529356$ as its other real root. Now try to confirm that around $h=0.11847271629$ chaotic things start to happen.
Actually, it looks like at that step size the "natural" equilibrium becomes unstable and the solution converges to an artificial equilibrium that depends on the step size. The switch to oscillating and later chaotic behavior happens at a slightly larger step size.

Plotting the oscillations like a Feigenbaum diagram (which is related to the Euler method for this same task) gives a typical period-doubling and then increasingly chaotic behavior. Now it depends on what kind of stability you are after.

