Ramanujan Style Nested Differential Equation

If the operator is meant to derive what follows, then we have $$y(x) lim_n to infty n! cdot y^(n)(x) qquad,qquad forall x in X$$since the derivative of a constant is always $0$ , and the derivative of a sum is the sum of derivatives. However, if multiplication is meant, with the last term of the nested product presumably being none other than y(x), then we have $$y(x) sum_n1^infty n! cdot y^(n)(x) qquad,qquad forall x in X$$where $y : X to Y$ ; either way, since $$lim_n to inftyn! infty$$ then, in order for the function to converge $forall x in X$ , we must have $$lim_n to infty y^(n)(x) 0 quad,quad forall x in X quad>quad y(x) P_m(x) sum_k0^m a_k cdot x^k quad,quad m in mathbbN$$since the Nth nested integral of $0$ is nothing else than a polynomial function of degree N-$1$

So I was exploring some math the other day... and I came across the following neat identity:

Given $y$ is a function of $x$ ($y(x)$) and

$$

y 1 fracmathrmdmathrmdx left(1 fracmathrmdmathrmdx left(1 fracmathrmdmathrmdx left(1 fracmathrmdmathrmdx left(1 fracmathrmdmathrmdx left( cdots right) right) right) right) right) text (repeated differential)

$$

then we can solve this equation as follows:

$$

y - 1 fracmathrmdmathrmdx left(1 fracmathrmdmathrmdx left( cdots right) right) iff int y - 1 , mathrmd x 1 fracmathrmdmathrmdx left( 1 fracmathrmdmathrmdx left( cdots right) right)

$$

$$

implies int y - 1 , mathrmd x y iff y - 1 fracmathrmd y mathrmd x

$$

So

$$

ln left( y - 1 right) x C iff y Ce^x 1

$$

This problem reminded me a lot of nested radical expressions such as:

$$

x 1 sqrt1 sqrt 1 sqrt cdots iff x - 1 sqrt1 sqrt 1 sqrt cdots

$$

$$

implies (x - 1)^2 x iff x^2 - 3x 1 0

$$

and so

$$

x frac32 fracsqrt52

$$

This reminded of the Ramanujan nested radical which is:

$$

x 0 sqrt 1 2 sqrt 1 3 sqrt1 4 sqrt cdots

$$

whose solution cannot be done by simple series manipulations but requires knowledge of general formula found by algebraically manipulating the binomial theorem...

This made me curious...

say $y$ is a function of $x$ ($y(x)$) and

$$

y 0 fracmathrmdmathrmdx left(1 2fracmathrmdmathrmdx left(1 3fracmathrmdmathrmdx left(1 4fracmathrmdmathrmdx left(1 5fracmathrmdmathrmdx left( cdots right) right) right) right) right)

$$

What would the solution come out to be?

Ramanujan Style Nested Differential Equation 1

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