Logit functions as solutions of differential equations

Question

Do you know if there exists a set of differential equations for which the solutions are logit functions?

If yes, are you aware of articles describing them?

Thank you very much in advance,

Answer 1

To be honest I cannot name you book or a source which could answer your question directly. Nevertheless I was able to construct an DE which solutions are given by Logit Functions.

Consider the Logit Function given by

$$\operatorname{logit}(x)=f(x)=\log\left(\frac{x}{1-x}\right)$$

I will suppose that we are working with the natural logarithm to keep things easy. Taking the first and the second derivative yields to

$$f'(x)=-\frac{1}{x(x-1)}~\text{and}~f''(x)=\frac{2x-1}{x^2(x-1)^2}=\frac2{x-1}\frac{1}{x(x-1)}-\left(\frac1{x(x-1)}\right)^2$$

Putting this together we can deduce a relation between the first and the second derivative. Namely

$$\frac2{x-1}f'(x)-(f'(x))^2=f''(x)$$

which is in fact a non-linear second order DE and has the general solution

$$f(x)=\log\left(c_1-\frac{c_2}{x-1}\right)$$

This can be done by the substitution $f'(x)=y$ and solving the new first order DE.