You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'The class of nonlinear ordinary differential equations $y''''y=F(z,y^2)$, where $F$ is a smooth function, is studied. Various nonlinear ordinary differential equations, whose applicative importance is well known, belong to such a class of nonlinear ordinary differential equations. Indeed, the Emden-Fowler equation, the Ermakov-Pinney equation and the generalized Ermakov equations are among them. Bäcklund transformations and auto Bäcklund transformations are constructed: these last transformations induce the construction of a ladder of new solutions adimitted by the given differential equations starting from a trivial solutions. Notably, the highly nonlinear structure of this class of nonlinear ordinary differential equations implies that numerical methods are very difficulty to apply.'
---

[ ]{}

[ ]{}

[**Keywords**]{}: Non-linear ordinary differential equations, Bäcklund transformations,\
Schwartzian derivative, Ermakov-Pinney equation, Emden-Fowler equation.

Introduction {#intro}
============

Since the 1990s, a number of results on discretization of ordinary differential equations describing integrable physical systems were achieved. Among them there are the Ruijsenaars-Schneider model [@Nijhoff], the Henon-Heiles, Garnier and Neumann systems [@HKR], [@Tsi1], the Ermakov-Pinney equation [@AH], the Euler top [@Bob], the Lagrange [@KPR] and Kirchoff tops [@Rag2] , the Chaplygin ball [@Tsi2], the Gaudin systems [@Rag1], [@FZ2] (see also [@FZ1] and references therein). All these discretizations represent maps among solutions of the differential equations describing the corresponding integrable systems. Usually, these transformations are obtained from the Lax representation of the model, by constructing an appropriate [*dressing matrix*]{} intertwining the Lax representation of the system (see e.g. [@FZ1] for details). These transformations turn out to be canonical in the phase space of the system. Furthermore, they enjoy some crucial properties inherited by the Lax representation and by the integrable structure of the system. In particular the maps are algebraic, and, if the Lax matrix is of order $N$, this algebraic structure enters through an auxiliary variable $\gamma$ which satisfies an irreducible polynomial equation of degree $N$ whose coefficients are rational functions of the dynamical variables (see e.g. [@HRZ]).\
In this work an algebraic map among solutions of a class of nonlinear second order differential equations is obtained. Integrability of the equations is not requested, neither the transformations are obtained via the Lax representation of the system. Rather, both the differential equations considered and the transformations obtained are an extension of the results given in [@JMAA2000]: in that work the properties of the Schwarzian derivative were fundamental to achieve the results. In the present case, the Schwarzian derivative continues to play a key role in the construction of the transformations, allowing us to derive new results.

The present study concerns non-linear ordinary differential equations of second order of the form $$\label{eq}
y y''=F(z,y^2)~, ~~y: {{\mbox{R\hskip-0.9em{}I \ }}}^+\to {{\mbox{R\hskip-0.9em{}I \ }}}^+ ,$$ where, as usual, the prime sign denotes derivative with respect to the independent variable and $F$ is a suitably regular given function of its arguments. A large number of differential equation which model physical phenomena can be written in the form (\[eq\]): we can mention the Ermakov-Pinney equation [@Common-Musette],[@Ermakov] and the Emden-Fowler equation (see e.g. [@Cha] and [@GH]). These two equations and their generalisations are further discussed in the subsequent sections. Finally, note that, when the function $F$ in (\[eq\]) depends on the independent $z$ variable only, equation (\[eq\]) reduces to the ordinary differential equation (1.1) in [@JMAA2000], which takes its physical origin in extended kinetic theory (see [@JMAA2000] and references therein).

It is well known, see [@Hille] for instance, that the Schwarzian derivative plays a fundamental role in the theory of *linear* second order differential equations. Given any smooth enough function $f(z)$ defined on an open set $I\subset{{\mbox{R\hskip-0.9em{}I \ }}}$, its Schwarzian derivative $\{f,z\}$ is defined via $$\label{Schw}
\{f,z\}:=\frac{f'''}{f'}-\frac{3}{2}\left(\frac{f''}{f'}\right)^2, ~~~ f'(z) \neq0~ ~~\forall z\in I.$$ The link between the Schwarzian derivative and the theory of linear differential equations is given by the following result (see e.g. [@Hille], Theorem 10.1.1): if $g_0$ and $g_1$ are two independent solutions of the equation $$\label{linear}
g''=2P' g'+Q g,$$ where $P$ and $Q$ denote smooth functions, say $C^2$, of the independent variable, then the Schwarzian derivative of the ratio $g_0/g_1$, assuming $g_1\neq 0$, depends only on the functions $P$ and $Q$. Specifically, it turns out $$\label{pr}
\{\frac{g_0}{g_1},z\}=2(P''-P'^2-Q).$$ The latter connects equations (\[linear\]) and (\[eq\]) via the Schwarzian derivative. In the following, Bäcklund transformations, admitted by (\[eq\]) itself, are obtained. Then, new explicit solutions of (\[eq\]) are constructed via Bäcklund transformations admitted by (\[eq\]) itself.

The material is organised as follows. The opening Section 2 is concerned about the construction of a Bäcklund transformation needed to establish the subsequent results. In particular, conditions which imply it is an auto-Bäcklund transformation are established. The functional equation which guaranties the Bäcklund transformation is an auto-Bäcklund transformation are studied in Section 3. In the following Section 4, the relation between the linear equation (\[linear\]), the equation (\[pr\]) and the auto-Bäcklund of the ordinary differential equations under investigation are considered. Notably, they exhibit terms in which the Schwarzian derivative appears.

Section 5 is devoted to show how some well known nonlinear ordinary differential equations are amenable to be treated via the method devised in the previous Sections. Specifically, the physically relevant cases of the Ermakov-Pinney and of the Emden-Fowler equation are studied in two different subsections where corresponding Bäcklund and auto Bäcklund transformations are constructed.

In the closing Section 6, further to some conclusions, also perspective investigations are mentioned.

Bäcklund transformations
========================

In this Section an invariance shown in [@JMAA2000] is revisited and suitably generalised to adapt to the wider class of nonlinear ordinary differential equations under investigation. To start with, let us introduce the following Lemma.

\[lemma1\] [Consider the ordinary differential equation , where $y: {{\mbox{R\hskip-0.9em{}I \ }}}^+\to {{\mbox{R\hskip-0.9em{}I \ }}}^+$, then admitted solutions are related, via Bäcklund transformation, to solutions of the ordinary differential equation $$\label{2}
v'' v -\frac{1}{2}(v')^2=2v F(z, v).$$ ]{}

**Proof** Apply the transformation $$\label{v}
T: ~~~~ v= y^2~~, ~~ y, v: {{\mbox{R\hskip-0.9em{}I \ }}}^+\to {{\mbox{R\hskip-0.9em{}I \ }}}^+,$$ and note that the kernel of $T$ is empty since $y$ and $v$ are assumed to be real positive valued. Then, direct substitution of in gives .

$\square$

The link can be graphically depicted as follows $$\boxed{y''y=F(z,y^2)}{\buildrel T \over{-\!-}}\boxed{v'' v -\frac{1}{2}(v')^2=2v F(z, v)}$$

\[lemma2\] [Consider the reciprocal transformation $$\label{reciprocal}
R:  ~~~~ \left\{\begin{array}{ccc}x = \Phi &  {D_t} = v {D_x},  \\ \\
  v =  D_t \Phi  & ~~~~~ D_x = [ \Phi_t]^{-1}{D_t}  \end{array}\right.$$ wherein: $$\label{R33}
\displaystyle{ {D_t} := {d  \over
dt},~~~~ {D_x} :=  {d  \over dx}, ~~~ \Phi_t = {d  \over dt} \Phi};$$ it transforms , where $v: {{\mbox{R\hskip-0.9em{}I \ }}}^+\to {{\mbox{R\hskip-0.9em{}I \ }}}^+$, in $$\label{eqphi}
\{\phi,t\}=2\phi_t F(\phi,\phi_t).$$ ]{}

**Proof** Direct application of the reciprocal transformation $R$ produces the result.

$\square$

The two Lemmata can be summarised in the following Bäcklund chart, according to the terminology in [@SIGMA2016][^1], where by [*Bäcklund chart*]{} is meant the net of links, represented by Bäcklund Transformations, among different equations. $$\boxed{y''y=F(z,y^2)}{\buildrel T \over{-\!-}}\boxed{v'' v -\frac{1}{2}(v')^2=2v F(z, v)}{\buildrel R \over{-\!-}}\boxed{\{\phi,t\}=2\phi_t F(\phi,\phi_t)}$$ Hence, a correspondence between equations (\[eq\]) and (\[eqphi\]), we term [*Schwarzian*]{} equation, is established. This result, combined with known properties of the Schwarzian derivative, allows to prove further invariances enjoyed by (\[eq\]).

\[prop2\] [Let $g$ and $\psi$ denote two suitably smooth maps, then assume $\phi$ can be expressed via the composition of such two maps, namely $\phi=g(\psi)$, then $\psi$ satisfies a Schwarzian equation of the same form of (\[eqphi\]), that is $$\label{eqphi2}
\{\psi,t\}=2\psi_t \tilde F(\psi,\psi_t)~,~ ~~\text{where}~ \tilde F~ \text{is a smooth function}.$$]{}

**Proof** Computation of the Schwarzian derivative $\{\phi,t\}$, where $\phi=g(\psi)$, composition of the two differentiable maps $g$ and $\psi$, gives $$\label{Schwarzcomp}
\{\phi,t\}=\psi_t^2\{g,\psi\}+\{\psi,t\}.$$ The latter, substituted in equation (\[eqphi\]) can be re-written as $$\{\psi,t\}=2\phi_tF(\phi,\phi_t)-\psi_t^2\{g,\psi\}.$$ Note that this equation is of the same form of (\[eqphi\]). Indeed, the derivative $\phi_t$ is $ g'\psi_t$, and hence $$\label{psi}
\{\psi,t\}=2g'\psi_tF(g,g'\psi_t)-\psi_t^2\{g,\psi\}.$$ Thus, if we define the new function $\tilde{F}$ via $$\tilde{F}(\psi,\psi_t):= g'F(g,g'\psi_t)-\frac{\psi_t}{2}\{g,\psi\},$$ and substitute it into equation (\[psi\]), the thesis readily follows since equation (\[eqphi2\]), of the form of (\[eqphi\]), is obtained. $\square$

Now, the Bäcklund chart can be extended setting $B_1: \phi-g(\psi)=0$, so that the links are summarised as follows. $${\small{\boxed{y''y=F(z,y^2)}{\buildrel T \over{-\!-}}\boxed{v'' v -\frac{1}{2}(v')^2=2v F(z, v)}{\buildrel R \over{-\!-}}\boxed{\{\phi,t\}=2\phi_t F(\phi,\phi_t) }{\buildrel {B_1} \over{-\!-}}\boxed{\{\psi,t\}=2\psi_t \tilde F(\psi,\psi_t) }}}$$

Accordingly, solutions admitted by the differential equation (\[eq\]) can be constructed as stated in the following proposition.

\[prop3\] [Let $f$ denote a smooth enough function such that $f'(z)\neq 0$, $\forall z\in{{\mbox{R\hskip-0.9em{}I \ }}}^+$ introduce $$\label{map}
Y^2(z):=\frac{y^2(f(z))}{f'(z)},$$ where $y(z)$ is any solution of the equation (\[eq\]), then $Y(z)$ is a solution of the equation $$\label{te}
YY''=f'F(f,f'Y^2)-\frac{1}{2}\{f,z\}Y^2~.$$ ]{}

**Proof** Note that can be regarded as the Bäcklund transformation: $$\label{map2}
B_2: ~~{f'(z)} Y^2(z)-{y^2(f(z))}=0$$ which, given the smooth function $f$, connects to . Thus, esplicit computations prove the result.

$\square$

Let us discuss this result. The transformation (\[map\]) represents a map between corresponding solutions of equations (\[eq\]) and (\[te\]). Two different cases can occur.\
That is, when the two equations (\[te\]) and (\[eq\]) exhibit different forms, the Bäcklund transformation $B_2$, given by (\[map2\]), connects solutions of equation (\[eq\]) with solutions of equation (\[te\]).\
A special case arises when equations (\[eq\]) and (\[te\]) share the same form: in this case, $B_2$, given by (\[map\]), represents an auto-Bäcklund transformation between solutions of equation (\[eq\]). The necessary and sufficient condition $B_2$ must satisfy to be an auto-Bäcklund transformation reads $$\label{fde}
F(z,v)=f'F(f,f'v)-\frac{1}{2}\{f,z\}v.$$ The latter is obtained via direct comparison between the r.h.s.s of the two equations (\[te\]) and (\[eq\]). This result is stated in the following corollary.

\[cor1\] Let $y(z)$ and $Y(z)$ be, in turn, solutions of the differential equations and (\[te\]), then $B_2$, in , represents an auto-Bäcklund transformation, whenever $f$ denotes a solution of the functional differential equation .

Equation is a functional differential equation: a characterisation of the solutions it admits is provided in the next section. Then, explicit examples are considered to show how it is possible to construct new solutions of equation (\[eq\]) on application of the transformation (\[map2\]).

The functional differential equation
====================================

This Section is concerned about the study of the functional equation (\[fde\]) wherein the unknown is the function $f$, while $F$ is given. Thus, it is a 3rd order nonlinear ordinary differential equation subject to the condition $f^{\prime}\neq 0$. Here, for the sake of simplicity, we do not look for $f$, which is a solution of (\[fde\]), but we prefer to consider (\[fde\]) as a linear non homogeneous ordinary differential equation in the unknown $F$, where, in addition, $F$ is assumed to depended on $v$ via a [*suitable*]{} power expansion. This approach allows us to say which forms the given function $F$ may assume in order to be amenable to the presented method. Specifically, given a suitable $F$, a method to construct solutions of equation , via Bäcklund transformations, is provided.

Accordingly, let us assume that the function $F(z,v)$ admits a formal power expansion in $v$, that is $$\label{Fexp}
F(z,v)=\sum_{n}F_{n}(z)v^n,$$ where the sum is intended on a suitable set of integers. For the sake of convenience, when $n\neq 1$, the coefficients $F_n(z)$ are looked for under the form $F_{n}=(G'_n)^{n+1}$, where $G'_n$ denotes the derivative of a smooth (at least $C^1$) function of $z$. In the case $n=1$, we set $$\label{F1}
F_1(z):=(G'_1)^{2}-\frac{1}{2}Q(z)$$ where $Q(z)$ is a suitable function. Hence, substitution of the latter and of in (\[fde\]), gives $$\label{fde1}
2\sum_{n}\left((f')(G'_n)(f)-(G'_n)(z)\right)^{n+1}v^n-(f')^{2}Q(f)v+Q(z)v-\{f,z\}v=0.$$ which depends explicitly on $G_n$, so that, for example, $(G'_n)(f)$ indicates that $G'_n$ is composed with $f(z)$.\
The power series expansion in (\[fde1\]) represents a polynomial in $v$ whose first term multiplies $v^{-1}$, hence equality (\[fde1\]) is satisfied when all the coefficients of $v^{k}$ are set equal to zero. This means that the action of the map $f(z)$ on the functions $G_n(z)$ is a translation, that is, the functions $G_n(z)$ satisfy the linear functional equations $$\label{G_nK}
G_n(f(z))=G_n(z)+K_n~~~\Longleftrightarrow~f'G'_n(f)=G'_n(z)~~\forall n,$$ where $K_n$ are arbitrary constants. Substitution of in (\[fde1\]) gives $$\label{fde2}
Q(z)=(f')^{2}Q(f)+\{f,z\}.$$ Notably, the right hand side of the latter reminds the right hand side of equation (\[Schwarzcomp\]) when, in turn, the variable $z$ is identified with the variable $t$ and the function $f$ with the function $\psi$. This correspondence suggests to identify $Q$ with a suitable Schwarzian derivative, that is, to set $$\label{position}
Q(z):=\{w,z\}.$$ Substitution of this position in , and the subsequent comparison of with (\[Schwarzcomp\]) show that solutions of are obtained whenever $w(z)$ and $f(z)$ are related via the functional equation $$\label{position2}  
\{w(z),z\}=(f')^{2}\{w(f),f\}+\{f,z\}, $$ wherein the Schwarzian derivative of the composition of two functions appears since $(f')^{2}\{w(f),f\}+\{f,z\}\equiv \{w(f),z\}$. Accordingly, the function $w(f(z))$ and $w(z)$ follow to be related via a fractional linear transformation, i.e. $$\label{flt}
w(f(z))=\frac{aw(z)+b}{cw(z)+d}, \qquad ad-bc\neq 0.$$ Hence, combination of the shown results, allows to prove the following proposition.

\[prop1\] If the function $F(z,v)$ can be represented as a power series expansion in $v$, then the solution of the functional differential equation $$F(z,v)=f'F(f,f'v)-\frac{1}{2}\{f,z\}v.$$ is given by $$\label{F}
F(z,v)=\sum_n a_n \left(G'_n (z)\right)^{n+1}v^n-\frac{1}{2}\{w(z),z\}v$$ where the functions $G_n(z)$ and $w(z)$ satisfy the functional equations ($a$, $b$, $c$, $d$ and $K_n$ are arbitrary constants) $$\label{functionaleqs}
G_n(f(z))=G_n(z)+K_n, \quad w(f(z))=\frac{aw(z)+b}{cw(z)+d}, \qquad ad-bc\neq 0.$$

The previous proposition allows to find differential equations possessing the auto-Bäcklund transformations (\[map2\]). A corollary of this proposition reads as follows.

\[cor1\] Let $y_0(z)$ be a solution of the differential equation $$\label{diffeq}
y''=\sum_n a_n \left(G'_n (z)\right)^{n+1}y^{2n-1}-\frac{1}{2}\{w(z),z\}y,$$ where the functions $G_n(z)$ and $w(z)$ are specified in Proposition \[prop1\], then $$y_1(z)^2=\frac{y_0(f(z))^2}{f'(z)}$$ represents an auto-Bäcklund transformation admitted by equation (\[diffeq\]).

These results can be enriched and complemented thanks to the following two propositions.

\[propnG\] The general solution of the functional equation $$\label{fe1}
G(f(z))=G(z)+K$$ is given by $$\label{sfe1}
G(z)=G_0(z)+\Phi(G_0(z))$$ where $G_0(z)$ is a particular solution of equation (\[fe1\]) and $\Phi(z)$ is an arbitrary periodic function of period $K$.

\[propnw\] Let $w(z)$ be defined as $$\label{sfe2}
w(z):=\frac{DG(z)-B}{A-CG(z)}~, ~~  A, B, C, D \in {{\mbox{R\hskip-0.9em{}I \ }}}$$ where $G(z)$ is any function satisfying equation (\[fe1\]), then $w(z)$ is a solution of $$\label{fe2}
w(f(z))=\frac{(\Delta+DCK)w(z)+D^2K}{(\Delta-DCK)-w(z)C^2K}.$$ where $\Delta = AD-BC\neq0$.

These propositions can be easily proved via direct computations.

Auto-Bäcklund transformations 
==============================

In this Section auto-Bäcklund transformations are related to linear second order ordinary differential equations. Specifically, the attention is focussed on how solutions admitted by second order *linear* ordinary differential equations lead to the construction of the transformation (\[map2\]).

First of all, the general ideas are presented. Consider the link between two independent solutions admitted by (\[linear\]) and let $w(z)$ denote their ratio, namely $$\label{w}
w(z):= \frac{g_0(z)}{g_1(z)}, ~~ g_1(z)\neq 0.$$ Then, according to (\[pr\]), the Schwarzian derivative of $w$ is given by $$-\frac{1}{2}\{w,z\}=Q+(P')^2-P'',$$ where $P$ and $Q$ are the smooth coefficients of the ordinary differential equation (\[eq\]). Assume, now, a function $f(z)$ exists such that $w(f(z))$ can be expressed as a fractional linear transformation of $w(z)$, i.e. it is such that it satisfies the hypotheses of proposition \[prop1\]. Consider, then, the fractional linear transformation: $$\label{wf}
w(f(z))=\frac{(\Delta+DCK)w(z)+D^2K}{(\Delta-DCK)-w(z)C^2K}~, ~~ 
    A, B, C, D, K \in {{\mbox{R\hskip-0.9em{}I \ }}}$$ where $\Delta=AD-BC\neq0$. Let the function $G(z)$ be defined via $$G(z):=\frac{Aw(z)+B}{Cw(z)+D},$$ then, the functional relation $G(f(z))=G(z)+K$ holds, $\forall z$. Note that the function $w(z)$ is given by (\[w\]); hence, the derivative of the function $G(z)$ can be written, on use of the Wronskian of solutions of equation (\[linear\]), as $$\label{Gprime}
G'(z)=\Delta C_1 \left(\frac{e^{P(z)}}{Cg_0(z)+Dg_1(z)}\right)^2,$$ where $C_1$ is a suitably chosen constant. When the arbitrary constants, denoted as $a_n$, are inserted as coefficients, the following, consequence of Corollary \[cor1\] is proved.

\[cor2\] Let $y_0(z)$ be a solution of the differential equation $$\label{diffeq1}
y''=\sum_n a_n \left(\frac{e^{P(z)}}{Cg_0(z)+Dg_1(z)}\right)^{2n+2}y^{2n-1}+\left(Q+(P')^2-P''\right)y,$$ where the functions $P(z)$ and $Q(z)$ are arbitrary and $g_0$ and $g_1$ are two independent solutions of the linear differential equation (\[linear\]). Then, if it is possible to find a function $f(z)$ such that equation (\[wf\]) holds, the map $$\label{BTs}
y_1(z)^2=\frac{y_0(f(z))^2}{f'(z)}$$ defines an auto-Bäcklund transformation admitted by equation (\[diffeq1\]).

The relevance of this result is stressed in the next section where the results (\[diffeq1\])-(\[BTs\]) are applied to different examples.

Applications
============

The class of differential equations (\[diffeq\]) can be shown to embrace, as particular cases, a large number of nonlinear differential equations. The aim of this Section is to show implications on the well known Ermakov-Pinney and Emden-Fowler nonlinear equations whose applicative relevance is undoubtable. In addition, some generalisations of these equations are also considered.

First of all, a simple case is treated. Note that if, in (\[diffeq\]), here re-written for convenience $$y''=\sum_n a_n \left(G'_n (z)\right)^{n+1}y^{2n-1}-\frac{1}{2}\{w(z),z\}y, \eqno(29)$$ we set $G_n(z):=G(z), ~\forall n$, it is compatible with the further positions $K_n = K$ and $\Phi =0$. Indeed, in the arbitrariness in the choice of the functions $G_n(z)$ is reflected in the arbitrariness of the parameters $K_n$ and of the periodic function $\Phi$, respectively, in equation (\[G\_nK\]) and (\[sfe1\]). Furthermore, the (tacitly) assumed convergence, in a given open set $\Omega\subset{{\mbox{R\hskip-0.9em{}I \ }}}$, of the series in (\[diffeq\]), implies that the sequences $\{a_n\}\in {{\mbox{R\hskip-0.9em{}I \ }}}$ is such that, in $\Omega\subset{{\mbox{R\hskip-0.9em{}I \ }}}$, the following convergent series can be introduced $$\label{H}
H(z):=\sum_{n} a_n z^n.$$ The Corollary \[cor1\] takes, in this case, the following simplified form.

\[cor4\] Let $y_0(z)$ be a solution of the differential equation $$\label{diffeq3}
yy''=G'H(G'y^2)-\frac{1}{2}\{w,z\}y^2,$$ where the functions $G(z)$ and $w(z)$, as specified in Proposition \[prop1\], satisfy the functional equations , that is $$G(f(z))=G(z)+K, \quad w(f(z))=\frac{aw(z)+b}{cw(z)+d}, ~~~a,b,c,d, K \in{{\mbox{R\hskip-0.9em{}I \ }}}.  \hskip3cm
(\ref{functionaleqs})$$ Then, the map $$y_1(z)^2=\frac{y_0(f(z))^2}{f'(z)}$$ defines an auto-Bäcklund transformation of the equation (\[diffeq3\]).

In the next subsections, differential equations with relevant physical meaning are obtained corresponding to [*ad hoc*]{} choices, in (\[diffeq3\]), of the function $H(z)$.

The Ermakov-Pinney equation
---------------------------

A first remarkable case is represented by the Ermakov-Pinney equation studied in this subsection. Specifically, if, in the equation (\[diffeq1\]), the function $P(z)$ as well as all the coefficients $a_n$ are all set equal to zero except $a_{-1}$, denoted via $a_1=-\alpha$, then, the Ermakov-Pinney equation [^2]: $$\label{Pinney}
y''=Q(z)y-\frac{\alpha}{y^3}$$ is obtained. This is equivalent to choose $H(z)=-\frac{\alpha}{z}$ in corollary \[cor4\].

The Ermakov-Pinney equation was introduced in 1880 by Ermakov [@Ermakov], who investigated solvability conditions of second order ordinary differential equations. The Ermakov-Pinney equation is closely connected to the harmonic oscillator with a time-dependent frequency. It is used to describe various problems, in quantum mechanics, such as the motion of charged particles in the Paul trap [@MGW], in atomic transport theory [@T] and Bose-Einstein condensate theory [@Nicolin]. Furthermore, it plays a fundamental role in the description of the unitary evolution of quantum non- autonomous systems [@VV]. It appears also in cosmology [@HL]. A set of Bäcklund transformation and an exact discretization admitted by equation (\[Pinney\]) is due to A. Hone [@AH], who, on application of two different transformations, provides its exact discretization.\
Now, on applications of results in the previous sections, a set of Bäcklund transformations admitted by the Ermakov-Pinney equation are obtained. In particular, since equation (\[Pinney\]) corresponds to $P=0$ in equation (\[diffeq1\]), the admitted Bäcklund transformations are specified by the following proposition.

\[PinneyBT\] Suppose that $y_0$ is a solution of the Ermakov-Pinney equation. Define the function $\displaystyle{w(z)=\frac{g_0}{g_1}}$, where $g_0$ and $g_1$ are two independent solutions of the linear differential equation $y''=Qy$. Then, if it is possible to find a function $f(z)$ such that equation (\[flt\]) holds, the map $$y_1(z)^2=\frac{y_0(f(z))^2}{f'(z)}$$ is an auto-Bäcklund transformation of the equation (\[Pinney\]).

**Proof** The proposition follows on specialisation, in Corollary \[cor2\], of all the parameters via $a_k=-\alpha\,\delta_{k,-1}$ and $ P\equiv 0$.

$\square$

Notable examples are represented by the case when the function $Q$ exhibits a dependence on $z^2$: in particular, it is a linear combination of a term proportional to $z^{-2}$ with a term proportional to $z^{4k}$ for some integer $k\in{{\mbox{N\hskip-0.9em{}I \ }}}$. Both these functions, taken separately, have application in scalar field cosmologies (see [@HL], eqs. (10)-(11)-(31)-(38)). Specifically, consider $Q$ to be $$\label{ex1}
Q(z)=p^2(2k+1)^2\frac{z^{4k}}{4}+\frac{k(k+1)}{z^2} ~~, ~ p\in{{\mbox{R\hskip-0.9em{}I \ }}}^+.$$ In this case, the function $w$ is obtained $$\label{wex1}
w(z)=e^{pz^{2k+1}}$$ and the function $f(z)$ can be defined by $$\label{fex1}
f^{2k+1}=\frac{1}{p}\ln\left(\frac{aw(z)+b}{cw(z)+d}\right)~~, ~ ad-bc\neq0.$$ Hence, given a solution $y_0$ of equation (\[Pinney\]), further solutions are represented by $$\label{BTex1}
y_1(z)^2=\frac{y_0(f)^2}{f'}.$$ Indeed, chosen $Q$ in (\[ex1\]), a particular solution of equation (\[Pinney\]) is given by $$\label{psex1}
y_0(z)=\frac{\beta}{z^k}, \qquad \beta^4:= \frac{4\alpha}{p^2(2k+1)^2}.$$ Thus, on application of the transformation (\[BTex1\]), after few manipulations, we find a new set of solutions, say $y_1(z)$, as follows $$\label{solex1}
y_1(z)^2=\frac{\beta^2}{z^{2k}}\frac{(aw(z)+b)(cw(z)+d)}{w(z)(ad-bc)}~~, ~ ad-bc\neq0~,$$ where $w(z)$ is defined by equation (\[wex1\]). Note that the previous equation defines the general solution of equation (\[Pinney\]) which corresponds to $Q$ in (\[ex1\]).

The Emden-Fowler equation
-------------------------

This subsection is devoted to a second example of an ordinary differential equation which admits Bäcklund transformations whose construction is possible according to the general results comprised in Sections 2-4. The generalised Emden-Fowler equation of the first kind in the unknown $q(x)$ reads (see e.g [@GH]) $$\label{EF1}
xq_{xx}+\alpha q_{x}+\beta x^{\nu}q^n=0.$$ This equation appears in a wide variety of mathematical physics problems: specifically, when $\nu=1$ and $\alpha=3$ it describes the hydrostatic and thermodynamic equilibrium of stars. In addition, it appears in investigations on the Einstein’s field equations [@Hertl] or in the mean-field description of critical adsorption [@GR] (see [@GH] and references therein for further examples). When the parameter $\alpha$ is integer, equation (\[EF1\]) is used to model spherically symmetric steady state solutions of evolution problems involving the Laplace operator in a $\alpha$-dimensional space. More precisely, when $r$ denotes the [*radial*]{} coordinate, looking for spherically symmetric solutions, the reduction of the equation $$\nabla^2 q+\beta r^{\nu-1}q^n=0$$ directly gives equation (\[EF1\]). A particularly interesting generalisation is represented by the modified Emden-Fowler equation, which is obtained on substitution of the factor $x^{\nu}$, in (\[EF1\]), with a function of $x$: $$\label{EFM}
xq_{xx}+\alpha q_x+\beta r(x)q^n=0.$$ In this section two examples are provided: one for equation (\[EF1\]) and the other for equation (\[EFM\]). Equation (\[EF1\]) is the particular case of equation (\[EFM\]), which corresponds to the choice $r(x)=x^{\nu}$. Likewise, our first example is a particular case of the second one.\
**Example 1**. In this example, let $\nu=1-2\alpha$ in (\[EF1\]) and, for later convenience, set $n=2m-1$. The differential equation (\[EF1\]) then reads $$\label{qeq}
x q_{xx}+\alpha q_x +\beta x^{1-2\alpha} q^{2m-1}=0.$$ The following change of variables $$q(x)=\frac{y(z)}{z}, \qquad z=x^{\alpha-1}$$ in (\[qeq\]) reduces to $$\label{EF1R}
y''+\frac{\beta}{(\alpha-1)^2} \frac{y^{2m-1}}{z^{2m+2}}=0.$$ The latter can be compared with equation (\[diffeq1\]): that is, according to Corollary \[cor2\], when we set $Q=P''-(P')^2$ and $a_m=- \frac{\beta}{(\alpha-1)^2}\delta_{m,n}$, equation (\[EF1R\]) is obtained. Indeed, the choice $Q=P''-(P')^2$ entails that the functions $g_0$ and $g_1$ are linear combinations of $e^P$ and $ze^P$ (independent solutions of equation (\[linear\])). Let, now chose $$g_0=e^P(D-bz),\qquad g_1=e^P(az-C), \qquad aD-bC=1$$ then, equation (\[diffeq1\]) reads exactly as (\[EF1R\]). The next step is to show, solving the functional equation (\[fe2\]), that the function $f(z)$ is given by $$f(z)=\frac{\Delta z}{Kz+\Delta},$$ which allows to find the Bäcklund transformations $$\label{BTexx1}
y_1^2(z)=y_0^2\left(\frac{\Delta z}{Kz+\Delta}\right)\frac{(Kz+\Delta)^2}{\Delta^2}.$$ A particular solution of equation (\[EF1\]) is given in [@GH]: it corresponds to the case when $$\label{firstsol}
{\displaystyle{y_0(z)=p \, z^{\frac{m}{m-1}} }}$$ and represents a solution of equation (\[EF1R\]), where the constant $p$ is required to satisfy $$\beta (m-1)^2p^{2m-2}+m(\alpha-1)^2=0.$$ Note that the latter, in general, given real $\beta$ and $m$, admits complex solutions $p\in{{\mbox{C\hskip-0.5em{}I \ }}}$; hence, to obtain $p\in{{\mbox{R\hskip-0.9em{}I \ }}}$, $\beta$ and $m$ are required to be, in turn, positive and negative, or viceversa. The new solution (\[BTexx1\]) reads $$\label{firstbac}
y_1^2(z)= p^2\left(\frac{\Delta z}{Kz+\Delta}\right)^{\frac{2m}{m-1}}\frac{(Kz+\Delta)^2}{\Delta^2}$$ and represents two different one-parameter solutions of equation (\[EF1R\]), the parameter we can chose is $\frac{K}{\Delta}$. Note also that further iterations of the same transformation, where each iteration is identified by the pair of parameters $\Delta_n$ and $K_n$, do not add further constants to the solution. Indeed, the $n^{th}$ iteration can be written as $$y_n^2(z)=p^2  \left(\frac{R_n z}{S_nz+R_n}\right)^{\frac{2m}{m-1}}\frac{(S_n z+R_n)^2}{R_n^2}$$ where the coefficients $R_n$ and $S_n$ solve the recurrence relations $$R_{n+1}=\Delta_{n+1}R_n,\quad S_{n+1}=\Delta_{n+1}S_n+K_{n+1}R_n$$ with the initial values $R_0=1$, $S_0=0$ (and obviously $\Delta_1=\Delta$ and $K_1=K$).\
**Example 2**. In this example we look at the modified Emden-Fowler equation (\[EFM\]). Again, for later convenience, we set $n=2m-1$. We specify the function $r(x)$ as $$\label{setr}
r(x)=\frac{1}{x^{2\alpha}}\left(\frac{x^{\alpha-1}}{\eta+\gamma x^{\alpha-1}}\right)^{2m+2},$$ which corresponds to the following differential equation in the unkown $q(x)$ $$\label{EFMq}
xq_{xx}+\alpha q_x+\beta x^{1-2\alpha}\left(\frac{x^{\alpha -1}}{\eta+\gamma x^{\alpha-1}}\right)^{2m+2} q^{2m-1}=0.$$ When the special values $(\eta,\gamma)=(0,1)$ are chosen, (\[EFMq\]) reduces to (\[qeq\]) of the previous example. The subsequent change of variables in (\[EFMq\]) $$q(x)=\frac{y(z)}{z}, \qquad z=x^{\alpha-1},$$ gives $$\label{yr}
y''+\frac{\beta}{(\alpha-1)^2}\frac{y^{2m-1}}{(\eta+\gamma z)^{2m+2}}=0.$$ This equation can be compared with equation (\[diffeq1\]): according to Corollary \[cor2\], we must set $Q=P''-(P')^2$ and $a_m=- \frac{\beta}{(\alpha-1)^2}\delta_{m,n}$. Then, the functions $g_0$ and $g_1$ are linear combinations of $e^P$ and $ze^P$ (that is, they are two independent solutions of equation (\[linear\])). Let $$\left\{\begin{split}
g_0&=e^P(a_0 z+b_0),\\
g_1&=e^P(a_1z+b_1).
\end{split}\right.
\quad \textrm{with}\quad 
\left\{\begin{split}
Ca_0+Da_1=\gamma,\\
 Cb_0+Db_1=\eta,
\end{split}\right.$$ These choices allow to check that equation (\[diffeq1\]) reduces exactly to (\[yr\]). Then it can be shown that the function $f(z)$, solution of the functional equation (\[fe2\]), is given by $$f(z)=\frac{(K\eta\gamma +\Delta\delta)z+K\eta^2}{(\Delta\delta-K\eta\gamma)-K\gamma^2z}, \qquad \delta:= a_0b_1-a_1b_0.$$ According to Corollary \[cor2\], the Bäcklund transformations admitted by equation (\[yr\]) are $$\label{secondbac}
y_1(z)^2=y_0\left(\frac{(K\eta\gamma +\Delta\delta)z+K\eta^2}{(\Delta\delta-K\eta\gamma)-K\gamma^2z}\right)^2\left(\frac{(\Delta\delta-K\eta\gamma)-K\gamma^2z}{\Delta\delta}\right)^2.$$ A particular solution of (\[EFMq\]) is given by $$\label{secondsol}
y_0(z)=p(\eta+\gamma z)^{\frac{m}{m-1}}$$ where the constant $p$ represents a solution of $$\beta (m-1)^2p^{2m-2}+m\gamma^2 (\alpha-1)^2=0.$$ Note that the solution (\[secondsol\]) reduces to (\[firstsol\]) when $(\eta,\gamma)=(0,1)$ and, then, the transformation (\[secondbac\]) coincides with the transformation (\[firstbac\]).\
The new solution is given by $$y_1(z)^2=p^2\left(\frac{\Delta\delta(\eta+\gamma z)}{(\Delta\delta-K\eta\gamma)-K\gamma^2z}\right)^{\frac{2m}{m-1}}\left(\frac{(\Delta\delta-K\eta\gamma)-K\gamma^2z}{\Delta\delta}\right)^2$$

Conclusions
===========

A class of nonlinear differential equations of second order of the form $y y''=F(z,y^2)$, which represents a genuine generalisation of $y~ y^{\prime\prime} = S(x),$ in [@JMAA2000], is considered in this article. The studied equations are proved to admit Bäcklund transformations (\[map\]) which, in the general case, represent maps from solutions of a nonlinear differential equations of the form (\[eq\]) to solutions of another equation of the same class. Notably, these Bäcklund transformations comprise also, as special cases, auto-Bäcklund transformations. These special cases are investigated; thus, whenever $F(z,v)$ and $f(z)$ satisfy the differential functional equation (\[fde\]), it is shown that the map (\[map2\]) represents an auto Bäcklund transformations admitted by equation (\[eq\]). These auto Bäcklund transformations are algebraic and, indeed, in the complex domain, they establish multi-valued correspondences among solutions of the same differential equations; however, when restrictions to positive real valued functions are considered, one-to-one correspondences are obtained. In particular, in the very special case when $F$ depends only on $z$, i.e. $F(z,v)=S(z)$, and, furthermore, the function $f$ is a linear fractional transformation, i.e. $f(z)=(az+b)/(cz+d)$, the transformation in [@JMAA2000] is obtained as a particular case of (\[map2\]). In addition, a variety of applications of equation (\[eq\]) are provided: they represents only some of the examples in which the constructed Bäcklund transformations can be applied to obtain new results. Further systems of physical interest, which, for instance arise as reductions of partial differential equations, belong to the class of equations we investigate: in the present article we restrict our attention to some physically relevant examples. Further results, in particular as far as the structure of equation (\[eq\]) as well as its solutions in the complex domain are in progress. Also, currently under investigation are relations between the obtained Bäcklund transformations, integrability properties and, possibly, the Hamiltonian structure of the linked equations, aiming to specialise the results in [@PhysA166].

[**Acknowledgments**]{}

The financial support of GNFM-INdAM, INFN and <span style="font-variant:small-caps;">Sapienza</span> Università di Roma is gratefully acknowledged.

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[^1]: The interested reader is referred to [@SIGMA2016] and Ref.s therein to track early occurences of the term [*Bäcklund chart*]{}.

[^2]: On introduction of the Kronecker symbol ${\displaystyle{\delta_{k, l}:= \left\{ { \begin{array}{c}
        0 ~~ k\neq l \cr
        1~~ k = l \\ 
      \end{array} } \right.}}$, the condition on the coefficients $a_n$ can be written as $a_n= -\alpha \delta_{-1, n}$.