Linearizable Exact and Explicit Analytic Solutions for Riccati Differential Equations

Abstract

In this paper, linearizable exact and explicit analytic solutions for classical Riccati differential equations are investigated in the complex field $\mathbb{C}$ by locally reducing the equations to other q-difference-differential equations and constructing invertible holomorphic solutions of the q-difference-differential equations. We discuss the existence of invertible analytic solutions of the q-difference-differential equations in three cases: (i) $\left|q\right|\ne 1,$ (ii) $\left|q\right|=1$ but not a root of unity, and (iii) $\left|q\right|=1$ but a root of unity. Especially for case (ii), we deal with the q-difference-differential equations under the Brjuno condition, which is weaker than the Diophantine condition. Finally, we give an algorithm for solving the Riccati differential equations, which is helpful to obtain their approximate solutions.

1. Introduction

The Riccati equation

$${\displaystyle \frac{dy}{dx}}=P\left(x\right)y^2+Q\left(x\right)y+R\left(x\right),$$

(1)

where $P,$ Q and R are all analytic functions in a neighborhood of the origin, is a special type of nonlinear differential equation. Research on this equation has a long history dating at least to 1720 [1], with early contributions to its analysis being made by Euler, Abel and Liouville [2,3,4,5,6]. It plays a significant role not only in algebraic geometry and mathematical physics [7,8,9,10] but also finds significant applications in various fields, including automatic control, quantum physics, nonlinear optics, and financial engineering [11,12,13,14,15,16]. In many applications there is a pressing need to obtain the solutions of the above equation [17,18,19]. It is still meaningful to seek a special solution of Equation (1) with certain properties. In this paper, we are interested in a special class of analytic solutions (linearizable) of Equation (1) with analytic coefficients P, Q and analytic non-homogeneous term $R.$ We aim to seek some specific solutions to Equation (1) that exhibit behavior resembling that of the linear part in some sense.

For a non-zero complex number $q,$ let $y\left(x\right)$ be an analytic solution of (1) satisfying $y\left(0\right)=0$ and $y^{\prime }\left(0\right)=q.$ We say that $y\left(x\right)$ is analytically linearizable if there exists an analytic function $\phi \left(x\right)$ satisfying $\phi \left(0\right)=0$ and ${\phi }^{\prime }\left(0\right)\ne 0$ such that ${\phi }^{-1}\circ y\circ \phi \left(x\right)=qx,$ i.e., $y\left(x\right)$ is analytically conjugate to its linear part. In general, the analytic solution of Equation (1) may not be linearized at the fixed point, as we illustrate in the following examples. First of all, we consider the following two complex Riccati differential equations

$$\begin{array}{c}\hfill {\displaystyle \frac{dy}{dx}}=y^2-1.\end{array}$$

(2)

It is straightforward to verify that

$$\begin{array}{ccc}\hfill y\left(x\right)& =& -tanh\left(x\right)=-\left(x-{\displaystyle \frac{x^3}{3}}+{\displaystyle \frac{2x^5}{15}}-{\displaystyle \frac{17x^7}{315}}+{\displaystyle \frac{62x^9}{2835}}-\cdots \right)\hfill \\ & =& -{\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{2^{2n}(2^{2n}-1)B_{2n}}{\left(2n\right)!}}{x}^{2n-1},\hfill \end{array}$$

(3)

where $B_{2n}$ is Bernoulli number ($B_2={\displaystyle \frac{1}{6}},B_4=-{\displaystyle \frac{1}{30}},B_6={\displaystyle \frac{1}{42}},\cdots$) and is an analytic solution to the Equation (2), where $tanh\left(x\right):={\displaystyle \frac{e^x-e^{-x}}{e^x+e^{-x}}}.$ However, $y\left(x\right)=-tanh\left(x\right)$ cannot be linearized at the origin. Indeed, suppose that there exists an analytic function

$$\begin{array}{c}\hfill \phi \left(x\right)=b_{1}x+b_2{x}^2+b_3{x}^3+\dots ,b_1\ne 0\end{array}$$

(4)

such that ${\phi }^{-1}\left(y\left(\phi \left(x\right)\right)\right)=-x,$ i.e.

$$\begin{array}{c}\hfill -tanh\left(\phi \right(x\left)\right)=\phi (-x).\end{array}$$

(5)

By substituting (3) and (4) into (5) and comparing the coefficients of $x^3$, we obtain that

$$-b_3+{\displaystyle \frac{b_1^3}{3}}=-b_3.$$

This leads to $b_1=0,$ which contradicts the assumption that $b_1\ne 0.$ This shows that the solution $y\left(z\right)=-tanh\left(x\right)$ of $\left(E{q}_{-}\right)$ cannot be analytically linearized near the origin.

The previous example includes the cases where $q=-1.$ Now, let us examine another case with $q\ne 1.$ Consider the following complex Riccati differential equation

$$\begin{array}{c}\hfill {\displaystyle \frac{dy}{dx}}=y^2+q.\end{array}$$

It is easy to see that

$$\begin{array}{c}\hfill y\left(x\right)=\sqrt{q}tan\left(\sqrt{q}x\right)=qx+{\displaystyle \frac{q^2}{3}}{x}^3+{\displaystyle \frac{2q^3}{15}}{x}^5+\cdots \end{array}$$

(6)

is an analytic solution in a neighborhood of the origin. However, the solution cannot be analytically linearized near the origin. In fact, if the solution (6) is linearizable at the origin, then there exists a function $\phi \left(x\right)$ satisfying $\phi \left(0\right)=0$ and ${\phi }^{\prime }\left(0\right)\ne 0$ such that

$$\begin{array}{c}\hfill -\sqrt{q}tan\left(\sqrt{q}\phi \left(x\right)\right)=\phi \left(qx\right),q=e^{2\pi i/3}.\end{array}$$

(7)

By taking the derivative of both sides of Equation (7) with respect to x and substituting $x=0,$ we obtain that $-q{\phi }^{\prime }\left(0\right)=q{\phi }^{\prime }\left(0\right).$ This implies ${\phi }^{\prime }\left(0\right)=0,$ which contradicts the assumption ${\phi }^{\prime }\left(0\right)\ne 0$.

At this point, it is natural to ask: under what conditions can we ensure the existence of linearizable analytic solutions of Equation (1)?

This paper will explore this direction. Our main goal is to study the existence of linearizable analytic solutions of Equation (1). We will provide conditions to guarantee the existence of linearizable analytic solutions of Equation (1) by using the classical majorant series method and dealing with small-divisor problems in one-dimensional dynamical systems. More precisely, we prove the existence of linearizable analytic solutions for (1) by performing the transformation

$$\begin{array}{c}\hfill y\left(x\right)=\phi \left(q{\phi }^{-1}\left(x\right)\right)\end{array}$$

(8)

to reduce the equation locally to another q-difference-differential equation

$$\begin{array}{c}\hfill q{\phi }^{\prime }\left(qx\right)={\phi }^{\prime }\left(x\right)[P\left(\phi \left(x\right)\right){\phi }^2\left(qx\right)+Q\left(\phi \left(x\right)\right)\phi \left(qx\right)+R\left(\phi \left(x\right)\right)],\end{array}$$

(9)

called the auxiliary equation of (1), and by constructing solutions in uniformly convergent power series for (9).

Throughout this paper, we always assume the following conditions:

(H) Assume that $P,$ Q and R are all analytic functions in the open disk ${\Delta }_{\sigma }=\{x\in \mathbb{C}:|x|<\sigma \}$ and $R\left(0\right)=q\ne 0,1,$ and q satisfies

(C1) $0<\left|q\right|\ne 1;$ or

(C2) $\left|q\right|=1$ but not a root of the unity; i.e., $q=e^{2\pi i\theta }$, and $\theta \in \mathbb{R}\setminus \mathbb{Q}$ is a Brjuno number, i.e.,

$$\begin{array}{c}\hfill B\left(\theta \right)={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{log{q}_{n+1}}{q_n}}<+\infty ,\end{array}$$

(10)

where $\{p_n/q_n\}$ denotes the sequence of partial fractions of the continued fraction expansion of $\theta ,$ said to satisfy the Brjuno condition; or

(C3) $q=e^{2\pi \mathrm{i}t/s}$ for some integer $s\in \mathbb{N}$ with $t\in \mathbb{Z}\setminus \left\{0\right\},$ and $q\ne e^{2\pi \mathrm{i}\xi /v}$ for all $1\le v\le s-1$ and $\xi \in \mathbb{Z}\setminus \left\{0\right\}$.

2. Invertible Analytic Solution for Auxiliary Equation (9)

We denote by $\mathbb{C}\left[\right[x\left]\right]$ and $\mathbb{C}\left\{x\right\}$, respectively, the set of all formal power series in x with coefficients in $\mathbb{C}$ and the set of all power series in $\mathbb{C}\left[\right[x\left]\right]$ that have non-zero convergent radius. Let $x\mathbb{C}\left[\right[x\left]\right]$ denote the set of formal series $xg\left(x\right)$, where $g\left(x\right)\in \mathbb{C}\left[\right[x\left]\right].$ Similarly, let $x\mathbb{C}\left\{x\right\}$ denote the set of convergent series $xg\left(x\right),$ where $g\left(x\right)\in \mathbb{C}\left\{x\right\}$.

We define the sets for some constant $\tau \ne 0$

$$\begin{array}{ccc}\hfill {\mathcal{F}}_{\tau }& =& \{\phi :\phi \left(x\right)\in x\mathbb{C}\left[\left[x\right]\right]\mathrm{and}{\phi }^{\prime }\left(0\right)=\tau \ne 0\}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {\mathcal{K}}_{\tau }& =& \{\phi :\phi \left(x\right)\in x\mathbb{C}\left\{x\right\}\mathrm{and}{\phi }^{\prime }\left(0\right)=\tau \ne 0\}.\hfill \end{array}$$

We first show Equation (9) has a solution $\phi \left(x\right)\in {\mathcal{F}}_{\tau }.$ Let

$$\begin{array}{c}\hfill \phi \left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{b}_n{x}^n,b_1\ne 0.\end{array}$$

(11)

In view of (H), we assume that P, Q and R have the power series expressions

$$\begin{array}{c}\hfill P\left(x\right)={\displaystyle \sum _{n=0}^{\infty }}{P}_n{x}^n,Q\left(x\right)={\displaystyle \sum _{n=0}^{\infty }}{Q}_n{x}^n\mathrm{and}R\left(x\right)={\displaystyle \sum _{n=0}^{\infty }}{R}_n{x}^n.\end{array}$$

(12)

Substituting to the power series (11) and (12) in (9), we obtain equations for the coefficients of (11)

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(q-R_0)b_1=0,\hfill \\ \\ 2(q^2-R_0)b_2=(Q_{0}q+R_1)b_1^2,\hfill \\ \\ (n+1)(q^{n+1}-R_0)b_{n+1}=D(n,P_i,Q_i,R_i,q,b_1,\dots ,b_n; 0\le i\le n),n\ge 2,\hfill \end{array}\right.\end{array}$$

(13)

where $D(n,P_i,Q_i,R_i,q,b_1,\dots ,b_n;0\le i\le n)$ is a polynomial.

In view of $R_0=R\left(0\right)=q,$ the coefficient of $b_1$ is zero in the first equality of (13). Hence, for any $\tau \in \mathbb{C}\setminus \left\{0\right\}$, we may take $b_1=\tau$, and the third equality of (13) is reduced to

$$\begin{array}{c}\hfill q(n+1)(q^n-1)b_{n+1}=D(n,P_i,Q_i,R_i,q,b_1,\dots ,b_n; 0\le i\le n).\end{array}$$

(14)

Note that $q\ne 0,1,$ and we can uniquely determine the sequence ${\left\{b_n\right\}}_{n=1}^{\infty }$ by (13) recursively. This implies that the formal series (11) is a solution of (9) in ${\mathcal{F}}_{\tau }$. If the formal power series (11) is an actual solution to Equation (9), we need to prove the convergence of the power series (11) in a neighborhood of the origin, i.e., $\phi \left(x\right)\in {\mathcal{K}}_{\tau }.$ We will discuss it in two subsections.

2.1. The Case of $\left|q\right|\ne$ 1

Theorem 1. 

Let the conditions (H) and (C1) be satisfied. Then, for any $\tau \in \mathbb{C}\setminus \left\{0\right\}$, the auxiliary Equation (9) has a solution $\phi \left(x\right)\in {\mathcal{K}}_{\tau }.$

Proof. 

In view of the condition (H) and the Cauchy inequalities, for any $r\in (0, \sigma )$, there exists a positive number M such that

$$\begin{array}{c}\hfill \left|P_n\right|,\left|Q_n\right|,\left|R_n\right|\le {\displaystyle \frac{M}{r^n}},n\ge 1.\end{array}$$

(15)

Let power series (11) be a formal solution of Equation (9), where $b_1=\tau \in \mathbb{C}\setminus \left\{0\right\},$ $b_2={\displaystyle \frac{(Q_{0}q+R_1){\tau }^2}{2q(q-1)}}$ and ${\left\{b_n\right\}}_{n=3}^{\infty }$ is determined by (14).

Furthermore, we note that, for $0<\left|q\right|\ne 1,$

$${\displaystyle \frac{1}{q(n+1)(q^n-1)}}D(n,P_i,Q_i,R_i,q,b_1,\dots ,b_n; 0\le i\le n)$$

is a polynomial whose coefficients are uniformly bounded with respect to $n.$ From (14) and (15), we know that there exists a sufficiently large positive number T such that

$$\begin{array}{ccc}\hfill |b_{n+1}|& \le & T·B(n,M/r^i,|b_1|,\dots ,|b_n|; 0\le i\le n),n\ge 2,\hfill \end{array}$$

(16)

where $B(n,M/r^i,|b_1|,\dots ,|b_n|; 0\le i\le n)$ is a polynomial with nonnegative coefficients.

We define a power series

$$\begin{array}{c}\hfill G\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{G}_n{x}^n\end{array}$$

(17)

by $G_1=\left|\tau \right|,$ $G_2=T(1+{\displaystyle \frac{M}{r}}){\left|\tau \right|}^2$ and

$$\begin{array}{ccc}\hfill G_{n+1}& =& T·B(n,M/r^i,|G_1|,\dots ,|G_n|; 0\le i\le n),n\ge 2,\hfill \end{array}$$

and then it is easy to see that, by induction,

$$\begin{array}{c}\hfill |b_n|\le G_n,n=1,2,\cdots .\end{array}$$

(18)

This implies the series (17) is a majorant series of (11). We now show that the series (17) has a positive radius of convergence. In fact, it is easy to prove that $G\left(z\right)$ formally satisfies the following functional equation

$$\begin{array}{cc} & G\left(x\right)-\left|\tau \right|x-T[G^3\left(x\right)+G^3\left(x\right)W\left(G\left(x\right)\right)\hfill \\ & +G^2\left(x\right)W\left(G\left(x\right)\right)+G^2\left(x\right)+G\left(x\right)W\left(G\left(x\right)\right)]=0,\hfill \end{array}$$

(19)

where

$$\begin{array}{c}\hfill W\left(x\right):={\displaystyle \sum _{n=1}^{\infty }}{\displaystyle \frac{M}{r^n}}{x}^n.\end{array}$$

(20)

In addition, by the implicit function theorem, we can prove that there exists a positive $\rho$ such that Equation (19) has a unique analytic solution $G\left(x\right)$ with $G\left(0\right)=0$ and $G^{\prime }\left(0\right)=\left|\tau \right|$ in $U_{\rho }:=\{x:|x|<\rho \}.$ In view of (18), the formal power series (11) is at least analytic in $U_{\rho }.$ This completes the proof. □

2.2. The Case of $\left|q\right|=1$

In this subsection, we are going to work on the existence of analytic solutions in the set ${\mathcal{K}}_{\tau }$ to the auxiliary Equation (9) in the case of $\left|q\right|=1.$ This kind of situation, q on the unit circle ${\mathbb{T}}^1,$ is a more difficult case since the small divisor $q^n-1$ is involved. We study the existence of analytic solutions to the auxiliary Equation (9) under the Brjuno condition (see (10)), which is weaker than the Diophantine condition. For related works, we refer the reader to [20,21,22]. For q on the unit circle ${\mathbb{T}}^1,$ we will discuss two scenarios: whether q is a root of unity or not.

We first consider that the case of $\left|q\right|=1$ but not a root of the unity. In this case, we need to deal with ‘small divisor’ $|q^n-1|.$ We first recall some basic facts. Let ${\epsilon }_n=|q^{n-1}-1|$ and define the sequence ${\left\{{\delta }_n\right\}}_{n=1}^{\infty }$ by ${\delta }_1=1$ and

$$\begin{array}{c}\hfill {\delta }_n={\epsilon }_n^{-1}\underset{2\le k\le n}{max}\underset{l_1+\dots +l_k=n}{max}{\delta }_{l_1}\cdots {\delta }_{l_k},n\ge 2.\end{array}$$

(21)

We can then express ${\delta }_{n_j}$ in terms of ${\epsilon }_{n_j}^{-1}$ and ${\delta }_{n_{j^{\prime }}}$ with $n_{j^{\prime }}<n_j.$ Proceeding like this, we finally arrive at the following decomposition,

$$\begin{array}{c}\hfill {\delta }_n={\epsilon }_{l_0}^{-1}{\epsilon }_{l_1}^{-1}\cdots {\epsilon }_{l_{\eta }}^{-1},n\ge 2,\end{array}$$

(22)

where $l_0=n,n>l_1,\cdots ,l_{\eta }\ge 1.$ Denote by $N_p\left(n\right)$ the number of factors ${\epsilon }_{l_j}$ in ${\delta }_n$ with ${\epsilon }_{l_j}<{\displaystyle \frac{1}{2}}\omega \left(p\right),$ where

$$\omega \left(p\right)=\underset{1\le i\le p}{min}\left\{\right|q^i-1|,\},p\ge 1.$$

Note that $\omega \left(p\right)$ is non-increasing with respect to p and we are interested in the case where $\omega \left(p\right)$ goes to zero as p goes to infinity. Similar to the proof in [23,24] or [20], we can prove the following Brjuno’s Lemma:

Lemma 1 

(Brjuno). For $p\ge 1,$ we have

$$\begin{array}{c}\hfill N_p\left(n\right)\le \left\{\begin{array}{cc}0,\hfill & n\le p,\hfill \\ {\displaystyle \frac{2n}{p}}-1,\hfill & n>p.\hfill \end{array}\right.\end{array}$$

Lemma 2 

(Siegel). If $n>n_1$, ${\epsilon }_n<{\displaystyle \frac{1}{2}}\omega \left(p\right)$ and ${\epsilon }_{n_1}<{\displaystyle \frac{1}{2}}\omega \left(p\right)$, then $n-n_1\ge p$.

Proof. 

By the assumption and the definition of ${\epsilon }_n$, it follows that

$$\begin{array}{ccc}\hfill \omega \left(p\right)& >& {\epsilon }_n+{\epsilon }_{n_1}\hfill \\ & =& |q^{n-1}-1|+|q^{n_1-1}-1|\hfill \\ & \ge & |q^{n-1}-q^{n_1-1}|\hfill \\ & =& |q^{n-n_1}-1|\hfill \\ & \ge & \omega (n-n_1),\hfill \end{array}$$

which implies $p\le n-n_1$ by the monotonicity of $\omega \left(p\right)$. □

Finally, we have the following lemma.

Lemma 3. 

If the Brjuno condition (10) is satisfied, then

$$\begin{array}{c}\hfill \underset{n}{sup}{\displaystyle \frac{1}{n}}log{\delta }_n<\infty .\end{array}$$

Proof. 

Since the condition (10) is satisfied, there exists a strictly increasing sequence ${\left\{q_{\nu }\right\}}_{\nu \ge 0}$ of integers with $q_0=1$ such that

$$\sum _{\nu \ge 0}{q}_{\nu }^{-1}log\omega {\left(q_{\nu +1}\right)}^{-1}<\infty .$$

For the proof, see Brjuno [24].

By the decomposition (22), we have

$${\displaystyle \frac{1}{n}}log{\delta }_n={\displaystyle \sum _{\mu =0}^{\eta }}{\displaystyle \frac{1}{n}}log{\epsilon }_{l_{\mu }}^{-1}.$$

By the Lemma 1, it follows that

$$card\{0\le \mu \le \eta :{\displaystyle \frac{1}{2}}\omega \left(q_{\nu +1}\right)\le {\epsilon }_{l_{\mu }}<{\displaystyle \frac{1}{2}}\omega \left(q_{\nu }\right)\}\le 2{\displaystyle \frac{n}{q_{\nu }}}$$

for $\nu \ge 1$ (detailed proof can be found in [24]). From the definition of ${\delta }_n,$ it is easy to see that the number of factors ${\epsilon }_{l_{\mu }}$ is bounded by $2n-1,$ which follows directly from (21) by induction. Consequently,

$${\displaystyle \frac{1}{n}}log{\delta }_n\le 2\sum _{\nu \ge 0}{\displaystyle \frac{1}{q_{\nu }}}log2\omega {\left(q_{\nu +1}\right)}^{-1}\le 2\left(\sum _{\nu \ge 0}{\displaystyle \frac{1}{q_{\nu }}}log\omega {\left(q_{\nu +1}\right)}^{-1}+log2\sum _{\nu \ge 0}{q}_{\nu }^{-1}\right).$$

(23)

Since $\omega \left(p\right)$ tends to zero monotonically as p goes to infinity, we can choose some $\tilde{p}$ such that $1>\omega \left(p\right)$ for all $p>\tilde{p},$ and we get

$$\sum _{\nu \ge {\nu }_0}{q}_{\nu }^{-1}\le {\displaystyle \frac{1}{log\omega {\left(\tilde{p}\right)}^{-1}}}\sum _{\nu \ge {\nu }_0}{q}_{\nu }^{-1}log\omega {\left(q_{\nu +1}\right)}^{-1}.$$

where ${\nu }_0$ verifies the inequalities $q_{{\nu }_0-1}\le \tilde{p}<q_{{\nu }_0}.$ Thus both series on the right-hand side of (23) converge thanks to the Brjuno condition. Therefore,

$$\underset{n}{sup}{\displaystyle \frac{1}{n}}log{\delta }_n<\infty .$$

□

Theorem 2. 

Let the conditions (H) and (C2) be satisfied. Then, for any $\tau \in \mathbb{C}\setminus \left\{0\right\}$, the auxiliary Equation (9) has a solution $\phi \left(x\right)\in {\mathcal{K}}_{\tau }.$

Proof. 

Let power series (11) be a formal solution of Equation (9), where $b_1=\tau \in \mathbb{C}\setminus \left\{0\right\},$ $b_2={\displaystyle \frac{(Q_{0}q+R_1){\tau }^2}{2q(q-1)}}$ and ${\left\{b_n\right\}}_{n=3}^{\infty }$ is determined by (14). In addition, from (14), it is easy to see that there exists a positive number

$$\begin{array}{c}\hfill L\ge \left|{\displaystyle \frac{r(Q_{0}q+R_1)|q+1|}{2(r+M)\left|q\right|}}\right|\end{array}$$

(24)

such that

$$\begin{array}{ccc}\hfill |b_{n+1}|& \le & L·B(n,M/r^i,|b_1|,\dots ,|b_n|; 0\le i\le n),\hfill \end{array}$$

(25)

where $B(n,M/r^i,|b_1|,\dots ,|b_n|; 0\le i\le n)$ is defined as in (16).

To construct a convergent majorant series of the series (11), we consider the functional equation for an unknown function $H\left(z\right)$

$$\begin{array}{cc} & H\left(x\right)-\left|\tau \right|x-L[H^3\left(x\right)+H^3\left(x\right)W\left(H\left(x\right)\right)\hfill \\ & +H^2\left(x\right)W\left(H\left(x\right)\right)+H^2\left(x\right)+H\left(x\right)W\left(H\left(x\right)\right)]=0,\hfill \end{array}$$

(26)

where $W\left(x\right)$ is defined in (20).

Similar to the discussion of Equation (19), we know that the functional Equation (26) has a unique analytic solution of the form $H\left(z\right)={\sum }_{n=1}^{\infty }{H}_n{x}^n$ in a neighborhood of the origin, and there exists a constant $S>0$ such that

$$\begin{array}{c}\hfill H_n<S^n,n\ge 1.\end{array}$$

(27)

Substituting $H\left(x\right)={\sum }_{n=1}^{\infty }{H}_n{x}^n$ into (26) and equating coefficients of like terms, we see that the sequence ${\left\{H_n\right\}}_{n=1}^{\infty }$ satisfies the following recurrence relations: $H_1=\left|\tau \right|,$ $H_2=L(1+{\displaystyle \frac{M}{r}}){\left|\tau \right|}^2$ and

$$\begin{array}{cc} & H_{n+1}=L·B(n,M/r^i,H_1,\dots ,H_n; 0\le i\le n),n\ge 2.\hfill \end{array}$$

(28)

Now, we can deduce, by induction, that $\left|b_n\right| \le H_n{\delta }_n$ for $n\ge 1$. In fact, $\left|b_1\right|=\left|\tau \right|=H_1$ and

$$\left|b_2\right|=\left|{\displaystyle \frac{(Q_{0}q+R_1){\tau }^2}{2q(q-1)}}\right|\le L\left(1+{\displaystyle \frac{M}{r}}\right){\left|\tau \right|}^2·{\displaystyle \frac{1}{|q^2-1|}}=H_2{\delta }_2,$$

and here we use inequality (24) and ${\delta }_1=1.$ For inductive proof we assume that $\left|b_j\right| \le H_j{\delta }_j, 1\le j\le n.$ Then, making use of the Lemma 3, (25) and (28), we get

$$\begin{array}{ccc}\hfill |b_{n+1}|& \le & {\displaystyle \frac{L}{|q^n-1|}}B(n,M/r^i,H_1{\delta }_1,\dots ,H_n{\delta }_n; 0\le i\le n)\hfill \\ & \le & {\displaystyle \frac{L}{|q^n-1|}}\underset{2\le t\le n+1}{max}\underset{l_1+\cdots +l_t=n+1}{max}{\delta }_{l_1}\cdots {\delta }_{l_t}B(n,M/r^i,H_1,\dots ,H_n; 0\le i\le n)\hfill \\ & =& H_{n+1}{\delta }_{n+1}\hfill \end{array}$$

as required. In view of (27), we have

$$\left|b_n\right|\le S^n{\delta }_n,n\ge 1.$$

Note again that, considering Lemma 3, we have that

$$\underset{n\to \infty }{lim}sup\left(\right|b_n{\left|\right)}^{{\displaystyle \frac{1}{n}}}\le \underset{n\to \infty }{lim}sup{\left(S^n{\delta }_n\right)}^{{\displaystyle \frac{1}{n}}}\le S·(\underset{n\to \infty }{lim}sup{\delta }_n^{{\displaystyle \frac{1}{n}}}).$$

This implies that the convergence radius of (11) is at least ${(S·({lim}_{n\to \infty }{sup}_n{\delta }_n^{{\displaystyle \frac{1}{n}}}))}^{-1}.$ That is, there exists a neighborhood $\mathcal{D}$ of the origin such that (11) is convergent in $\mathcal{D}.$ □

Next, we consider the case that $\left|q\right|=1$ and q is a root of unity. In this situation the Brjuno condition is violated; this is the so-called resonant case. Let $s\in \mathbb{N}$ be the smallest positive integer such that $q^s=1$. Then $q=e^{2\pi it/s}$ with $t\in \mathbb{Z}\setminus \left\{0\right\}$, and $q\ne 1$. From the recurrence (13) we see that the coefficient $b_{n+1}$ satisfies (14). Since $q^n-1=0$ if and only if n is a multiple of s, the factor $q^n-1$ vanishes exactly when $n=sl$ ($l\in \mathbb{N}$). Consequently, for $n=sl$, the recurrence (14) does not determine $b_{sl+1}$ unless $D(sl,P_i,Q_i,R_i,q,b_1,\dots ,b_{sl}; 0\le i\le sl)$ also vanishes. This is precisely the resonance phenomenon in the sense of normal form theory: the numbers $n=sl$ correspond to resonances, and the quantities $D(sl,P_i,Q_i,R_i,q,b_1,\dots ,b_{sl}; 0\le i\le sl)$ are the resonant coefficients. If $D(sl,P_i,Q_i,R_i,q,b_1,\dots ,b_{sl}; 0\le i\le sl)=0$ for all l, the resonant terms can be eliminated and a formal conjugation to the linear part $qx$ exists; if $D(sl,P_i,Q_i,R_i,q,b_1,\dots ,b_{sl}; 0\le i\le sl)\ne 0$ for some l, the obstruction is permanent and no formal power series solution is possible. To study the analyticity of the formal solutions when all resonances vanish, we construct a majorant series. Let

$$\gamma :=max\left\{1,{\displaystyle \frac{1}{|q-1|}},{\displaystyle \frac{1}{|q^2-1|}},\dots ,{\displaystyle \frac{1}{|q^{s-1}-1|}}\right\}.$$

Then, for every n with $n\not\equiv 0\left(\mathrm{mod}s\right)$, we have ${|q^n-1|}^{-1}\le \gamma$. Choose a constant $L\ge \left|{\displaystyle \frac{r(Q_{0}q+R_1)|q+1|}{2(r+M)\left|q\right|\gamma }}\right|$ such that, for all $n\ge 2$ with $n\not\equiv 0\left(\mathrm{mod}s\right)$,

$$\begin{array}{c}\hfill |b_{n+1}|\le L\gamma ·B(n,M/r^i,|b_1|,\dots ,|b_n|; 0\le i\le n).\end{array}$$

(29)

Now, we consider the functional equation for an unknown function $V\left(x\right)={\sum }_{n=1}^{\infty }{V}_n{x}^n$:

$$\begin{array}{ccc}\hfill V\left(x\right)-\left|\tau \right|x& -& L\gamma [V^3\left(x\right)+V^3\left(x\right)W\left(V\left(x\right)\right)+V^2\left(x\right)W\left(V\left(x\right)\right)\hfill \\ & +& V^2\left(x\right)+V\left(x\right)W\left(V\left(x\right)\right)]=0,\hfill \end{array}$$

(30)

where $W\left(x\right)$ is defined in (20). By the implicit function theorem, Equation (30) possesses a unique analytic solution in a neighborhood of the origin. Substituting the power series $V\left(x\right)={\sum }_{n=1}^{\infty }{V}_n{x}^n$ into (30) and comparing coefficients, one finds that the sequence $\left\{V_n\right\}$ satisfies $V_1=\left|\tau \right|$, $V_2=L\gamma (1+M/r){\left|\tau \right|}^2$, and, for $n\ge 2,$

$$\begin{array}{ccc}\hfill V_{n+1}& =& L\gamma ·B(n,M/r^i,V_1,\dots ,V_n; 0\le i\le n).\hfill \end{array}$$

(31)

Theorem 3. 

Suppose that (H) and (C3) hold, and let the coefficients $\left\{b_n\right\}$ be defined by the recurrences (14) with $b_1=\tau \ne 0$ and $b_2={\displaystyle \frac{(Q_{0}q+R_1){\tau }^2}{2q(q-1)}}$.

(i) If $D(sl,P_i,Q_i,R_i,q,b_1,\dots ,b_{sl}; 0\le i\le sl)=0$ for all $l\in \mathbb{N}$, then, for any choice of complex numbers ${\eta }_{sl+1}$ satisfying $\left|{\eta }_{sl+1}\right|\le V_{sl+1}$ ($l\in \mathbb{N}$), the auxiliary Equation (9) possesses an analytic solution in ${\mathcal{K}}_{\tau }$ of the form

$$\varphi \left(x\right)=\tau x+{\displaystyle \frac{(Q_{0}q+R_1){\tau }^2}{2q(q-1)}}{x}^2+{\displaystyle \sum _{l=1}^{\infty }}{\eta }_{sl+1}{x}^{sl+1}+\sum _{n\ge 3,n\not\equiv 1\left(\mathrm{mod}s\right)}{b}_n{x}^n.$$

(32)

In particular, setting all ${\eta }_{sl+1}=0$ yields a concrete convergent series whose coefficients are uniquely determined by the recurrence (14) and the rule $b_{sl+1}=0$ for $n=sl$. This provides an explicit iterative algorithm to construct an analytic solution.

(ii) If $D(sl,P_i,Q_i,R_i,q,b_1,\dots ,b_{sl}; 0\le i\le sl)\ne 0$ for some $l\in \mathbb{N}$, then Equation (9) admits no formal power series solution and consequently no analytic solution.

Proof. 

(i) Assume $D(sl,P_i,Q_i,R_i,q,b_1,\dots ,b_{sl}; 0\le i\le sl)=0$ for all $l\in \mathbb{N}$. We define $\left\{b_n\right\}$ as follows: set $b_1=\tau$ and $b_2$ as given. For $n\ge 2$, if $n\not\equiv 0\left(\mathrm{mod}s\right)$, let $b_{n+1}$ be given by (14). If $n=sl$, then $q^n-1=0$ and $D(sl,P_i,Q_i,R_i,q,b_1,\dots ,b_{sl}; 0\le i\le sl)=0$, so Equation (14) is satisfied for any value of $b_{sl+1}$; we choose $b_{sl+1}={\eta }_{sl+1}$ with $\left|{\eta }_{sl+1}\right|\le V_{sl+1}$. (For the concrete solution, take ${\eta }_{sl+1}=0$.) We now verify by induction that $\left|b_n\right|\le V_n$ for all $n\ge 1$. For $n=1,2$, the inequality holds by the definitions of $V_1,V_2$ and the choice of L. Suppose $\left|b_j\right|\le V_j$ for all $1\le j\le n$. If $n+1=sl+1$ for some l, then $\left|b_{n+1}\right|=\left|{\eta }_{sl+1}\right|\le V_{sl+1}=V_{n+1}$ by our selection. If $n\not\equiv 0\left(\mathrm{mod}s\right)$, then, from (29), (31) and the induction hypothesis, we obtain

$$\begin{array}{ccc}\hfill |b_{n+1}|& \le & L\gamma ·B(n,M/r^i,|b_1|,\dots ,|b_n|; 0\le i\le n)\hfill \\ & \le & L\gamma ·B(n,M/r^i,V_1,\dots ,V_n; 0\le i\le n)\hfill \\ & =& V_{n+1}\hfill \end{array}$$

as required. This completes the induction. Since $V\left(x\right)$ is analytic near the origin, the power series $\sum V_n{x}^n$ has a positive radius of convergence, say $\rho >0$. The estimate $\left|b_n\right|\le V_n$ implies that the series (32) converges at least for $\left|x\right|<\rho$, so it defines an analytic function $\varphi \left(x\right)$ in a neighborhood of zero with $\varphi \left(0\right)=0$ and ${\varphi }^{\prime }\left(0\right)=\tau \ne 0$. By construction, it satisfies (9). Thus $\varphi \in {\mathcal{K}}_{\tau }$.

(ii) If $D(sl,P_i,Q_i,R_i,q,b_1,\dots ,b_{sl}; 0\le i\le sl)\ne 0$ for some l, then, for $n=sl$, the left-hand side of (14) is zero while the right-hand side is non-zero, which makes it impossible to find a coefficient $b_{sl+1}$ satisfying the recurrence. Hence no formal power series solution exists. □

Remark 1. 

The condition $D(sl,P_i,Q_i,R_i,q,b_1,\dots ,b_{sl}; 0\le i\le sl)=0$ is exactly the cancellation of the resonant terms that would otherwise obstruct the formal linearization of the Riccati equation. This is completely analogous to the classical Poincaré–Dulac normal form theory for differential equations in the complex domain. When some resonant coefficient does not vanish, one cannot conjugate the equation to its linear part; however, it may still be possible to conjugate it to a polynomial normal form containing the unavoidable resonant monomials. Such a conjugation would produce a formal (possibly divergent) series, or a genuine solution if one allows non-analytic transformations (e.g., involving exponentials of logarithmic terms in the spirit of the Bruno–Sternberg theory). The exploration of these more general solutions in the context of Riccati equations is an interesting problem that we leave for future work.

3. Main Results

3.1. Linearizable Analytic Solutions of (1)

Theorem 4. 

If $\phi \left(x\right)\in {\mathcal{K}}_{\tau },$ then $\phi \left(x\right)$ is a solution of the auxiliary Equation (9) if and only if $y\left(x\right)=\phi \left(q{\phi }^{-1}\left(x\right)\right)$ is a solution of Equation (1) in $x\mathbb{C}\left\{x\right\}$ and $y^{\prime }\left(0\right)=q.$

Proof. 

Let $\phi \left(x\right)$ be a solution of Equation (9) and belong to ${\mathcal{K}}_{\tau }.$ In view of $\phi \left(0\right)=0$ and ${\phi }^{\prime }\left(0\right)=\tau \ne 0,$ we know that $\phi$ has an analytic inverse function ${\phi }^{-1}\left(x\right)$ near $x=0.$ Making $y\left(x\right)=\phi \left(q{\phi }^{-1}\right)\left(x\right)$, it is easy to prove that $y\left(x\right)$ is an analytic solution of Equation (1) with $y\left(0\right)=0$ and $y^{\prime }\left(0\right)=q.$ In fact,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dy}{dx}}& =& {\displaystyle \frac{q\phi \left(q{\phi }^{-1}\left(x\right)\right)}{{\phi }^{\prime }\left({\phi }^{-1}\left(x\right)\right)}}=P\left(x\right){\left(\phi \left(q{\phi }^{-1}\left(x\right)\right)\right)}^2+Q\left(x\right)\phi \left(q{\phi }^{-1}\left(x\right)\right)+R\left(x\right)\hfill \\ & =& P\left(x\right)y^2+Q\left(x\right)y+R\left(x\right).\hfill \end{array}$$

and $y\left(0\right)=\phi \left(q{\phi }^{-1}\left(0\right)\right)=\phi \left(0\right)=0,$ $y^{\prime }\left(0\right)={\displaystyle \frac{q{\phi }^{\prime }\left(q{\phi }^{-1}\left(0\right)\right)}{{\phi }^{\prime }\left({\phi }^{-1}\left(0\right)\right)}}={\displaystyle \frac{q\tau }{\tau }}=q.$ Conversely, if $\phi \left(x\right)\in {\mathcal{K}}_{\tau }$ and let $y\left(x\right)=\phi \left(q{\phi }^{-1}\left(x\right)\right)$ be an analytic solution of Equation (1), then it is easy to see that $\phi \left(x\right)$ is a solution of (9). This completes the proof of the theorem. □

The following two corollaries can be derived from Theorem 4.

Corollary 1. 

If one of the conditions of Theorems 1–3 is fulfilled, then Equation (1) has an analytic solution of the form (8).

The above corollary demonstrates that Equation (1) possesses a linearizable analytical solution in a neighborhood of the origin under the conditions specified in Theorems 1, 2 or 3.

3.2. Construction of Exact and Explicit Analytic Solutions of (1)

In this subsection, we show how to explicitly construct a linearizable exact analytic solution of (1) by means of (13). Let $P\left(x\right),$ $Q\left(x\right)$ and $R\left(x\right)$ satisfy condition (H) and $R\left(0\right)=q$ be a complex number that satisfies one of (C1), (C2) and (C3). By means of Theorems 1–3, Equation (9) has an analytic solution of the form

$$\phi \left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{b}_n{x}^n,$$

where $b_1=\tau \ne 0,$ and ${\left\{b_n\right\}}_{n=2}^{\infty }$ is determined by (13). If we assume ${\phi }^{-1}\left(x\right)={\sum }_{n=1}^{\infty }{c}_n{x}^n,$ then we can deduce that

$$\begin{array}{c}\hfill \sum _{1\le t\le n,\left(l_j\right)\in {\mathcal{A}}_n^t}{b}_t{c}_{l_1}\dots c_{l_t}={\delta }_{n1}\end{array}$$

(33)

from $\phi \left({\phi }^{-1}\left(x\right)\right)=x,$ where

$${\delta }_{n1}=\left\{\begin{array}{c}1,\mathrm{as}n=1,\hfill \\ \\ 0,\mathrm{as}n\ne 1.\hfill \end{array}\right.$$

From (33) we get the sequence ${\left\{c_n\right\}}_{n=1}^{\infty }$ successively. Thus, the exact explicit solution of Equation (1),

$$\begin{array}{c}\hfill y\left(x\right)=\phi \left(q{\phi }^{-1}\left(x\right)\right)={\displaystyle \sum _{n=1}^{\infty }}\left(q^n\sum _{1\le t\le n,\left(l_j\right)\in {\mathcal{A}}_n^t}{b}_t{c}_{l_1}\dots c_{l_t}\right)x^n,\end{array}$$

is obtained, and it can be linearized. Here we use the Faà di Bruno formula and

$${\mathcal{A}}_n^t:=\{(l_1,l_2,\dots ,l_t)\in {\mathbb{Z}}^t:l_j>0,j=1,2,\dots ,t,l_1+l_2+\dots +l_t=n\}.$$

Actually, once the existence of the analytical solution satisfying $\phi \left(0\right)=0$ and ${\phi }^{\prime }\left(0\right)=\tau \ne 0$ to Equation (9) is guaranteed, we can derive

$$\begin{array}{ccc} & & {\phi }^{″}\left(0\right)={\displaystyle \frac{(Q\left(0\right)q+R^{\prime }\left(0\right)){\tau }^2}{q(q-1)}},\hfill \\ & & {\phi }^{‴}\left(0\right)={\tau }^3\left[{\displaystyle \frac{R^{″}\left(0\right)+q{Q}^{\prime }\left(0\right)+2q^{2}P\left(0\right)}{q(q^2-1)}}+{\displaystyle \frac{(Q\left(0\right)+\frac{R^{\prime }\left(0\right)}{q})(Q\left(0\right)(2+q)+\frac{3R^{\prime }\left(0\right)}{q})}{(q-1)(q^2-1)}}\right],\hfill \\ & & etc,\hfill \end{array}$$

by taking successive derivatives of both sides of Equation (9) instead of first finding from (13).

Next, recall from the proof of the above theorem that ${\phi }^{-1}\left(x\right)$ is analytic in a neighborhood of the point $\phi \left(0\right)=0.$ Therefore, it can also be determined once its derivatives at $x=0$ have been determined:

$$\begin{array}{ccc} & & {\left({\phi }^{-1}\right)}^{\prime }\left(0\right)={\displaystyle \frac{1}{{\phi }^{\prime }\left({\phi }^{-1}\left(0\right)\right)}}={\displaystyle \frac{1}{{\phi }^{\prime }\left(0\right)}}={\displaystyle \frac{1}{\tau }},\hfill \\ & & {\left({\phi }^{-1}\right)}^{″}\left(0\right)=-{\displaystyle \frac{{\phi }^{″}\left({\phi }^{-1}\left(0\right)\right){\left({\phi }^{-1}\right)}^{\prime }\left(0\right)}{{\left({\phi }^{\prime }\left({\phi }^{-1}\left(0\right)\right)\right)}^2}}=-{\displaystyle \frac{{\phi }^{″}\left(0\right){\left({\phi }^{-1}\right)}^{\prime }\left(0\right)}{{\left({\phi }^{\prime }\left(0\right)\right)}^2}}=-{\displaystyle \frac{(Q\left(0\right)q+R^{\prime }\left(0\right))}{q(q-1)\tau }},\hfill \\ & & {\left({\phi }^{-1}\right)}^{‴}\left(0\right)=-{\displaystyle \frac{\left\{{\phi }^{‴}\left({\phi }^{-1}\left(0\right)\right){\left[{\left({\phi }^{-1}\right)}^{\prime }\left(0\right)\right]}^2+{\phi }^{″}\left({\phi }^{-1}\left(0\right)\right){\left({\phi }^{-1}\right)}^{″}\left(0\right)\right\}{\left[{\phi }^{\prime }\left({\phi }^{-1}\left(0\right)\right)\right]}^2}{{\left[{\phi }^{\prime }\left({\phi }^{-1}\left(0\right)\right)\right]}^4}}\hfill \\ & & +{\displaystyle \frac{{\phi }^{″}\left({\phi }^{-1}\left(0\right)\right){\left({\phi }^{-1}\right)}^{\prime }\left(0\right)·2·{\phi }^{\prime }\left({\phi }^{-1}\left(0\right)\right){\phi }^{″}\left({\phi }^{-1}\left(0\right)\right){\left({\phi }^{-1}\right)}^{\prime }\left(0\right)}{{\left[{\phi }^{\prime }\left({\phi }^{-1}\left(0\right)\right)\right]}^4}}\hfill \\ & & =-{\displaystyle \frac{1}{\tau q^2{(q-1)}^2}}[{\displaystyle \frac{(qQ\left(0\right)+R^{\prime }\left(0\right))(Q\left(0\right)q(q+2)+3R^{\prime }\left(0\right))+(2P\left(0\right)q^2+2Q^{\prime }\left(0\right)q+R^{″}\left(0\right))q(q-1)}{q+1}}\hfill \\ & & -3{(qQ\left(0\right)+R^{\prime }\left(0\right))}^2],etc.\hfill \end{array}$$

Finally, we determine a solution $y\left(x\right)$ of (1) by finding its derivatives at $x=0:$

$$\begin{array}{ccc} & & y\left(0\right)=\phi \left(q{\phi }^{-1}\left(0\right)\right)=\phi (q·0)=0,\hfill \\ & & y^{\prime }\left(0\right)={\phi }^{\prime }\left(q{\phi }^{-1}\left(0\right)\right)·q{\left({\phi }^{-1}\right)}^{\prime }\left(0\right)=q{\phi }^{\prime }\left(0\right){\left({\phi }^{-1}\right)}^{\prime }\left(0\right)=q\tau ·{\displaystyle \frac{1}{\tau }}=q,\hfill \\ & & y^{″}\left(0\right)=qQ\left(0\right)+R^{\prime }\left(0\right),\hfill \\ & & y^{‴}\left(0\right)={\displaystyle \frac{(qQ\left(0\right)+R^{\prime }\left(0\right))(q(q+2)Q\left(0\right)+3R^{\prime }\left(0\right)-3{(qQ\left(0\right)+R^{\prime }\left(0\right))}^2)}{q(q-1)}}\hfill \\ & & +2q^{2}P\left(0\right)+2q{Q}^{\prime }\left(0\right)+R^{″}\left(0\right):=\beta ,etc.\hfill \end{array}$$

Thus, the desired solution is

$$y\left(x\right)=qx+{\displaystyle \frac{1}{2}}(qQ\left(0\right)+R^{\prime }\left(0\right))x^2+{\displaystyle \frac{\beta }{6}}{x}^3+\cdots .$$

It is clear from the above procedure that, with the help of commercial software that is capable of exact differentiations, an arbitrary number of terms of the above series can be obtained.

Remark 2. 

In this paper, we have established a systematic framework for obtaining linearizable exact and explicit analytic solutions to the classical Riccati differential equation

$${\displaystyle \frac{dy}{dx}}=P\left(x\right)y^2+Q\left(x\right)y+R\left(x\right)$$

in the complex domain. The central idea is the transformation $y\left(x\right)=\varphi \left(q{\varphi }^{-1}\left(x\right)\right)$ with $q=R\left(0\right)\ne 0,1$, which reduces the Riccati equation to an equivalent q-difference-differential Equation (9) for the unknown function ϕ. Several directions for future research emerge from this work:

1. 

Extension to matrix and operator Riccati equations. The transformation technique may be generalized to matrix Riccati equations arising in optimal control and filtering, where the linear part is replaced by a matrix multiplier.

2. 

Convergence acceleration and numerical schemes. The rate of convergence of the power series solution can be investigated, and Padé approximants or other summation methods could be employed to extend the domain of validity.

3. 

Global properties and singularities. The present analysis is local (near the origin). It would be valuable to study the analytic continuation of the solutions and the nature of their singularities in relation to the Brjuno condition.

4. 

Other nonlinear equations. The method of reducing a differential equation to a q-difference-differential equation via a linearization transformation may apply to other classes of nonlinear ODEs, such as Abel equations or certain Painlevé-type equations.

5. 

Weaker conditions than Brjuno. It is known that the Brjuno condition is sufficient but not necessary for the convergence of linearization in one-dimensional dynamics (the optimal condition is the Yoccoz condition). Whether the results of Theorem 2 can be strengthened to the optimal arithmetic condition is an open and challenging problem.

Finally, we present two examples to validate Theorems 1 and 2.

Example 1 

(The case $\left|q\right|\ne 1$). Consider the Riccati equation

$${\displaystyle \frac{dy}{dx}}=y^2+xy+{\displaystyle \frac{1}{2}},x\in \mathbb{C}.$$

(34)

Here $P\left(x\right)=1$, $Q\left(x\right)=x$, $R\left(x\right)=1/2$. Thus

$$P_0=1,P_n=0(n\ge 1);Q_0=0,Q_1=1,Q_n=0(n\ge 2);R_0=1/2,R_n=0(n\ge 1),$$

and $q=R_0=1/2$ satisfies condition (C1).

Choosing $\tau =1$, the recurrence (13) yields:

$$\begin{array}{ccc}\hfill b_1& =& 1,\hfill \\ \hfill b_2& =& {\displaystyle \frac{(Q_{0}q+R_1){\tau }^2}{2q(q-1)}}={\displaystyle \frac{0·\frac{1}{2}+0}{2·\frac{1}{2}·(-\frac{1}{2})}}=0,\hfill \\ \hfill b_3& =& {\displaystyle \frac{1}{3q(q^2-1)}}\left[P_0{b}_1^3{q}^3+Q_0{b}_1^2{q}^2\right]={\displaystyle \frac{1}{3·\frac{1}{2}·(-\frac{3}{4})}}·{\displaystyle \frac{1}{8}}=-{\displaystyle \frac{1}{9}},\hfill \end{array}$$

and so on. The first few terms of $\varphi \left(x\right)$ are

$$\varphi \left(x\right)=x-{\displaystyle \frac{1}{9}}{x}^3-{\displaystyle \frac{1}{36}}{x}^4+\cdots .$$

By the Lagrange inversion formula (or the recurrence (33)), the inverse series is

$${\varphi }^{-1}\left(x\right)=x+{\displaystyle \frac{1}{9}}{x}^3+{\displaystyle \frac{1}{36}}{x}^4+\cdots .$$

Finally, the solution $y\left(x\right)=\varphi \left(q{\varphi }^{-1}\left(x\right)\right)$ with $q=1/2$ is

$$\begin{array}{ccc}\hfill y\left(x\right)& =& {\displaystyle \frac{1}{2}}x-{\displaystyle \frac{1}{8}}{x}^2+{\displaystyle \frac{1}{96}}{x}^3-{\displaystyle \frac{1}{192}}{x}^4+\cdots \hfill \\ & =& 0.5x-0.125x^2+0.0104167x^3-0.0052083x^4+\cdots .\hfill \end{array}$$

One can verify by direct substitution into (34) that the power series satisfies the equation up to the computed order $O\left(x^5\right)$.

Example 2 

(The case $\left|q\right|=1$, not a root of unity). Consider

$${\displaystyle \frac{dy}{dx}}=y^2+q,q=e^{2\pi i\theta },\theta ={\displaystyle \frac{\sqrt{5}-1}{2}}\left(\mathrm{the}\mathrm{golden}\mathrm{ratio}\right).$$

(35)

Here $P\left(x\right)=1$, $Q\left(x\right)=0$, $R\left(x\right)=q$. The number θ is irrational and satisfies the Brjuno condition (10) (in fact, it satisfies the Diophantine condition). By Theorem 2 and Corollary 1, Equation (35) admits a linearizable analytic solution. Choosing $\tau =1$, the coefficients $b_n$ are determined by (14). The first few terms (computed symbolically) are

$$\varphi \left(x\right)=x+{\displaystyle \frac{q}{q-1}}{x}^2+{\displaystyle \frac{q(2q^2+q+2)}{3(q-1)(q^2-1)}}{x}^3+\cdots ,$$

and the corresponding solution $y\left(x\right)=\varphi \left(q{\varphi }^{-1}\left(x\right)\right)$ begins with

$$y\left(x\right)=qx+{\displaystyle \frac{q^2}{3}}{x}^3+{\displaystyle \frac{2q^3}{15}}{x}^5+\cdots ,$$

which matches the Taylor expansion of $\sqrt{q}tan\left(\sqrt{q}x\right)$ when $q\ne \pm 1$ and $\left|q\right|=1$, confirming that the method correctly recovers the known analytic solution in this special case.

These examples demonstrate that the theoretical existence results in Theorems 1–3 and Corollary 1 are computationally realizable, and that the power series constructed via (13) indeed converge to the desired analytic solutions.