Abstract
The first part of the paper presents a unique reduction of some general Riccati-type differential equation to the general Bernoulli-type equation. Furthermore four special theorems on reduction of some Riccati-type differential equations are considered in the second part of the paper. A number of examples are also presented to illustrate the considered issues.
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Introduction
The aim of the first part of the article is to describe a unique reduction effect of the general Riccati-type differential equation [1,2,3,4]. The originality of this effect consists in the uniqueness of solution of respective differential equation. This is perhaps the most surprising property of the presented reduction. Let us get to the merits of the discussed problem. In the paper, we show that if the function \(y_1\) is any particular solution of the differential equation
where \(L[\cdot ]\) is some linear differential operatorFootnote 1 and \(q, r \in C(I)\), \(I\subset \mathbb {R}\) is an open interval, then for every \(N\in \mathbb {N}\), \(N\ge 3\), there exists a uniquely determined polynomial \(P\in \mathbb {R}[y]\), \(\deg P=N\), \(P(0)=P'(0)=P''(0)=0\), such that the linear substitution \(y=\frac{N}{2}y_1+z\) reduces the differential equation
to the following general Bernoulli-type equation of the form
where \(a_N=\frac{(-1)^N}{N-1}\left( \frac{2}{N}\right) ^{N-1}\). We begin with a simple illustration of this fact.
Lemma 1
Assume that the function \(y_1\ne 0\) is a solution of the Bernoulli differential equation
Then, the substitution \(y=\frac{3}{2} y_1+z\) reduces the equation
to the Bernoulli equation.
Proof
We have
which implies
By the assumption, we have
therefore, from Eq. 1, we get the Bernoulli equation
Example 1
Let us consider the following equation
Since the function \(y\equiv 1\) is a solution of the differential equation \(y'+y=y^2\), the substitution \(y=\frac{3}{2} +z\) reduces Eq. 2 to the form
which possesses the solutions of the form
where \(C\in [0,\infty ).\)
Generalization
The following generalization of Lemma 1 holds.
Theorem 1
Let \(y_1\) be a particular solution of the differential equation
where \(p,q\in C(I)\) and \(I\subset \mathbb {R}\) is a non-empty and non-degenerate interval. If \(y_1\ne 0\), then, for every \(N\in \mathbb {N}\), \(N\ge 3\), the substitution \(y=\frac{N}{2}y_1+z\) reduces the following differential equation
where
to the Bernoulli equation of the form
Remark 1
We note that Eq. 4 implies the identity
for each \(k=1,\ldots , N-2\), \(a_2:=1\).
Sketch of the proof
The proof comes down to verification of the following identities for the coefficients \(a_i\)
for every \(k=1,\ldots , N-2\) and
Identities Eq. 6 result (after the substitution \(y=\frac{N}{2} y_1+z\) in Eq. 3 and the expansion of the powers) from the form of the coefficients at \(z^{N-k}\) for \(k=1,\ldots , N-2\), respectively, identity Eq. 7 corresponds to the coefficient at \(z^0\).
Notice that identity Eq. 6 is equivalent to the convolution identity
which after applying the equality
implies
which gives \((1-1)^k=0.\) Therefore, identity Eq. 6 is true.
Wherein, Eq. 7 is equivalent to the identity
that is
which finishes the proof.
The above theorem is also attractive because of the following fact.
Theorem 2
Under the assumptions of Theorem 1, if for some \(\beta \in \mathbb {R}\) the substitution \(y=\beta y_1+z\) reduces the following differential equation
where \(\alpha _k\in C(I)\), \(k=2,\ldots ,N\) and \(\alpha _2(x)=q(x)\) to the Bernoulli equation of the form
then the coefficients \(\beta\) and \(\alpha _k(x)\) for \(k=3,\ldots ,N\) are the same as in Theorem 1 that is \(\beta =\frac{N}{2}\) and
Proof
The statement results directly from the calculations presented in the early stage of the proof of Theorem 1 leading to the generating of formulae Eqs. 6 and 7.
Remark 2
It is worth pointing out that both Theorem 1 and Theorem 2 can be generalized to the case when the linear differential operator
is replaced by the general linear differential operator
where \(p_r\in C(I)\), for \(r=0,1,\ldots ,n\), \(n\in \mathbb {N}\), and, what is interesting, the right side of the respective equation remains the same as in Eq. 3.
There is one more possibility of generalization of Theorem 1. Namely, Theorem 1 can be generalized to the case of Riccati equations and even to the case of general Riccati equations.
Theorem 3
Let \(y_1\) be a particular solution of the following n-th order general Riccati equations
where \(p_s,q,r\in C(I)\), \(s=0,1,\ldots ,n\), and \(I\in \mathbb {R}\) is a certain non-empty and non-degenerate interval. If \(y_1\ne 0\) for \(x\in I\), then for every \(N\in \mathbb {N}\), \(N\ge 3\), the substitution \(y=\frac{N}{2}y_1+z\) reduces the differential equation of the form
where coefficients \(a_k\) are defined as in Theorem 1, to the following general Bernoulli-type equation
Proof
The proof is analogous to the proof of Theorem 1.
Example 2
Let \(y_1\) be a solution of the following classical Riccati equation
which possesses “two” special solutions \(y_1=\frac{1\pm \sqrt{1-4a^2}}{-2ax}\) - see Remark 3. Then, the substitution \(y=2y_1+z\) reduces the following differential equation
to the Bernoulli equation
Remark 3
(see [5]) The classical Riccati equations \(y'=a(y^2+x^n)\), where \(n\in \left\{ \frac{4k}{1-2k}:k\in \mathbb {Z}\right\} \cup \{-2\}\), which obey the Eq. 8, were solved by Daniel Bernoulli and Jacopo Riccati in 1724 by the substitution
which implies
and
Hence, if \(n=-2\) and we assume (the special assumption) that \(v(x)=-y(x)\), then by Eq. 9
i.e.
which are two special solutions of the differential equation
Some special Riccati equations
The authors were also interested in nonlinear substitutions in Riccati equations. In addition to the known results, for example, the fundamental fact that the Riccati equation can be rewritten as linear second order homogeneous differential equation, we were also interested in specific cases absent from the known literature, see [6,7,8,9,10]. We will present four original results.
Theorem 4
If almost canonical form of Riccati equation is separable differential equation
where \(\alpha , \beta \in C(I)\), \(\alpha (x)\ne 0\) for every \(x\in I\), \(I \subset \mathbb {R}\) is an interval, and
and a function f(x) is a solution of this differential equation, \(f(x)\ne 0\) for every \(x \in I\), then function:
is a solution of Riccati differential equation:
where \(a,b \in C(I)\), \(b(x) \ne 0\) for every \(x \in I\), whenever
Proof
Setting \(y=\frac{1}{f}-y_0\) we get:
and since \(\frac{a(x)}{b(x)}=-2y_0\), we get
\(\square\)
The inverse version of Theorem 4 also holds.
Theorem 5
If \(y_0\in \mathbb {R}\) and function y(x) is particular solution of Riccati differential equation
where \(a,b \in C(I)\), \(I \subset \mathbb {R}\) is an interval, \(b(x)\ne 0\) for every \(x\in I\), and
then function
is a solution of the following Riccati equation:
where \(\alpha , \beta \in C(I),\,\alpha (x)\ne 0\) for every \(x\in I\), whenever
This equation is then a separable differential equation.
Proof
It follows easily from proof of Theorem 4.
Theorem 6
Let \(a,b\in C(I), \, I \subset \mathbb {R}\) be an interval, \(y_0\in \mathbb {R}, \, \alpha (x):=\exp \left( \int (2y_0b(x)-a(x))dx\right)\) and let \(y\ne 0\) be a solution of the following almost canonical Riccati differential equation:
Then, the function:
is a solution of Bernoulli equation:
Proof
We have:
which implies:
We also find:
which, by the assumption on the form \(\alpha\), gives:
At the end of this section, we will present one more result, which we will illustrate with an appropriate example.
Theorem 7
Let \(a,b\in C(I),\,c,d\in C^1(I)\), where \(I\subset \mathbb {R}\) is a nonempty and nontrivial interval. If \(d(x)\not \equiv 0\) for every \(x\in I\), then the substitution \(y(x)=f(x)-\frac{c(x)}{3d(x)}\) reduces both differential equations:
and
to the Riccati equation and the Riccati equation of second order, respectively.
Proof
Substituting \(y(x)=f(x)-\frac{c(x)}{3d(x)}\) we find:
and
Example 3
The substitution \(y(x)=f(x)-\tan x\) reduces the following general–type Riccati equation:
to the Bernoulli differential equation:
which possesses the solutions:
where \(C\in \mathbb {R}\) and either \(x\in \mathbb {R}\) whenever \(C>\frac{1}{2}\) or \(\frac{1}{2}\cos 2x+C>0\) if \(C\in \left( -\frac{1}{2},\frac{1}{2}\right]\).
Data availability
On behalf of all authors, the corresponding author declares that the data supporting the findings of this study are available within the paper.
Notes
Let us consider the space of functions \(f:U\rightarrow \mathbb {R}\), where \(U\subset \mathbb {R}\) is a fixed open set. Then, differential operator L of order n on this space is defined as:
$$L(x)=\sum \limits _{i=1}^n\alpha _i(x)\frac{d^i}{dx^i},$$where \(\alpha _i\in C(U)\) for every \(i=1,\ldots ,n\).
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Hetmaniok, E., Różański, M., Smuda, A. et al. ON THE REDUCTION OF SOME GENERAL RICCATI-TYPE EQUATIONS. J Math Sci (2024). https://doi.org/10.1007/s10958-024-07120-1
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DOI: https://doi.org/10.1007/s10958-024-07120-1