On the integrability of the Abel and of the extended Li\'{e}nard equations

Man Kwong Mak; Tiberiu Harko

REVIEW 4 major objections 2 minor 52 references

On the integrability of the Abel and of the extended Li\'{e}nard equations

T0 review · 4 major / 2 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Extended Liénard equations are solved exactly via Abel equations with known particular solutions.

desk verdict Theorems 2 and 3 have checkable algebraic errors (conditions (52) and (56) do not follow from the preceding equations), so the paper is only conditionally useful until those are fixed. read the letter →

arxiv 1908.03730 v1 pith:ZBNJ5DFK submitted 2019-08-10 math.CA math-phmath.MPnlin.SI

Man Kwong Mak , Tiberiu Harko This is my paper · ORCID

classification math.CAmath-phmath.MPnlin.SI MSC 34A0534A2534B3034C1534G20

keywords AbelequationextendedLiénardChielliniintegrabilityconditiongeneralsolutionquadraturesRiccatinonlinearoscillationsexact

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes exact general solutions for a wide family of nonlinear second-order equations—the extended Liénard equation—by exploiting its equivalence to a first-order Abel equation. The key step is the substitution $v=1/y'$, which turns $y''+f(y)(y')^n+k(y)(y')^m+g(y)y'+h(y)=0$ into $\frac{dv}{dy}=f(y)v^{3-n}+k(y)v^{3-m}+g(y)v^2+h(y)v^3$. If one particular solution of this Abel equation is known and the coefficients satisfy a Chiellini-type differential condition, the general solution of the second-order equation can be written as a quadrature. This matters because many nonlinear oscillator models, including quadratic-cubic damping equations and reduced Riccati equations, fall under these hypotheses and become exactly solvable.

What carries the argument

The machinery is the reduction to Abel equations and the separation step it enables. Setting $v=1/y'$ converts the extended Liénard equation into $\frac{dv}{dy}=f(y)v^{3-n}+k(y)v^{3-m}+g(y)v^2+h(y)v^3$. For $n=2$, $m=3$, subtracting a known particular solution $v_p$ and writing $U=Ew$ with $E=\exp\int[f+2gv_p+3hv_p^2]\,dy$ turns the difference equation into $w'=(g+3hv_p)Ew^2+hE^2w^3$, a Chiellini-type Abel equation. The Chiellini integrability condition $\frac{d}{dy}\left[\frac{E}{v_p-v_{p0}}\right]=9Sh(v_p-v_{p0})E$, where $v_{p0}=-g/(3h)$, then makes this equation separable. For $g=h\equiv0$ and arbitrary exponents, the generalized Chiellini lemma substitutes $F=P(f/k)^{1/(\beta-\alpha)}$, reducing the generalized Abel equation to a separable equation for $\theta$ and leading to the quadrature solution of Theorem 4.

What would settle it

Take smooth functions $f,g,h$ satisfying condition (52), compute the right-hand side of (55) with arbitrary integration constant $C$, differentiate twice, and substitute into equation (24); the theorem is true for that coefficient triple only if the residual vanishes for both sign choices. Likewise, testing the Abel solution (53) in (25) under the same condition must give an identity. One explicit triple where either residual fails would disprove the corresponding claim.

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Extended reading notes

Core claim

The paper claims that the extended Liénard equation $y''+f(y)(y')^n+k(y)(y')^m+g(y)y'+h(y)=0$ is exactly solvable, in closed form or by quadratures, whenever its associated Abel equation has one known particular solution and the coefficients satisfy a differential condition of Chiellini type. For the quadratic-cubic case $n=2$, $m=3$, three integrability classes are presented: the case $v_p=-g/(3h)$ with condition (37) and solution (38); the case governed by condition (52), whose Abel solution is (53)--(54) and whose Liénard solution is the quadrature (55); and the case of an arbitrary known $v_p$, where $f(y)$ is forced by (56) and the general solution is (63) with $\theta$ from (59). For $g=h\equiv0$ and arbitrary $n\neq m$, the generalized Chiellini condition (78)--(79) yields the general Liénard solution (80)--(82), and the reduced Riccati equation is integrated explicitly under condition (84).

Load-bearing premise

All of the solution formulas presuppose that a particular solution $v_p$ of the associated Abel equation is already known, and in Theorem 2 the coefficient $k(y)$ must be the one that actually makes $v_p$ a solution of (26); if such a $v_p$ cannot be found, the construction does not start.

Editorial extensions

If this is right

For the quadratic-cubic extended Liénard equation, coefficients satisfying condition (52) give the general solution as the quadrature (55), so such nonlinear oscillators are solved without numerical integration.
When a particular solution of the associated Abel equation is known and $f(y)$ satisfies condition (56), the general solution follows from the two quadratures (59) and (63).
For $g=h\equiv0$ and arbitrary $n\neq m$, coefficient pairs satisfying (78) or (79) yield the general solution via the single quadrature (80)--(82).
The reduced Riccati equation is exactly integrable in closed form under the condition $\frac{d}{dy}\sqrt{f/k}=Kf$, with solutions (85)--(86).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The integrability conditions are differential constraints on the coefficients, so they describe special, not generic, Liénard equations; the resulting solutions are natural benchmarks for numerical integrators and perturbation methods.
The paper does not give a constructive method for finding the particular solution $v_p$; a systematic way to generate such $v_p$ for coefficient families would turn these conditional results into a decision procedure for exact solvability.
For the Riccati case, condition (84) may be equivalent to a known solvable class under the standard transformation $v=u'/u$; checking that equivalence could place the new case inside a wider integrability hierarchy.
The solution in Theorem 2 also depends on solving the algebraic equation (54) for $\theta$, so practical use will require selecting the correct branch of the inverse function.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's the short version: the paper has a solid first integrability case and a plausible generalized Chiellini lemma, but two of the three quadratic-cubic theorems contain algebraic conditions that do not follow from the preceding equations. I would not trust the advertised formulas until those are corrected.

What is actually new: the setup is standard—transform y'' + f(y)(y')^2 + k(y)(y')^3 + g(y)y' + h(y) = 0 to an Abel equation for v = 1/y'. Theorem 1, where vp = -g/(3h), gives a correct quadrature formula under condition (37); that part checks out. The genuinely new content is the extension of the Chiellini condition to Abel equations with g = h = 0 and arbitrary exponents n, m (Lemma 2 and Theorem 4), plus the Riccati reduction in Theorem 5. Those look formally sound, though I did not verify every line.

The soft spots are in the middle. Theorem 2(a) states condition (52) as if it follows from Eq. (51). It does not. Integrating (51) gives B = 1/(18S∫h dy + C) with B = g^2/(9h^2) - f/(3h), i.e. a reciprocal relation; (52) is linear in ∫h dy. So Theorem 2's condition is wrong. Also, the theorem says a condition on f, g, h suffices, but the Abel equation contains k; the particular solution vp in (47) must satisfy Eq. (26), which ties down k. That consistency condition is not stated.

Theorem 3 has a similar, more directly checkable error. The Chiellini condition (44) with E = e^{∫F} and Q = g/h + 3vp reduces to F = Q'/Q + S h Q^2. Since F = f + 2gvp + 3hvp^2, the correct condition on f is f = Q'/Q + S h Q^2 - 2gvp - 3hvp^2. The printed Eq. (56) instead has (ln Q)'' and a derivative of hQ^2, which is not equivalent. A concrete example: h = 1, g = y, vp = 0, S = 1 gives f = y^2 + 1/y from the corrected condition (and then (44) holds), while (56) gives f = -1/y^2 + 2y, for which (44) fails. So formula (63) is unsupported as written.

Net: the first and last parts of the paper are worth reading, but the central quadratic-cubic results are currently unreliable. The errors are algebraic and probably fixable. I would send this to peer review, not desk-reject, but any referee should be told to re-derive conditions (52) and (56) and check the k-consistency. After that, the paper might be acceptable.

Referee Report

4 major / 2 minor

Summary. The paper studies the extended Liénard equation (21) via the transformation v=1/y' to an associated first-kind Abel-type equation. It claims closed-form general solutions in three cases for the quadratic-cubic Liénard equation, an arbitrary-particular-solution quadrature formula (Theorem 3), a generalized Chiellini integrability condition for the case g=h=0 (Lemma 2 and Theorem 4), and a Chiellini-type integrability condition for the reduced Riccati equation (Theorem 5). The method is standard: transform to Abel, assume a particular solution, then use the Chiellini condition to reduce to separable equations and quadratures.

Significance. If the claims are correct, the paper would provide a useful and explicit collection of integrability conditions and solution formulas for generalized Liénard equations. The generalized Chiellini lemma and the Riccati application appear algebraically sound, and the formulas are explicit enough to be checked by direct substitution. The paper is also transparent about its main limitation, namely that the existence of a particular solution of the associated Abel equation is assumed. However, the central theorem for arbitrary particular solutions contains a concrete algebraic error, and Theorem 2 as stated omits a necessary consistency condition on the coefficient k. These issues affect the main claims and require correction before the paper can be relied upon.

major comments (4)

[II.B, Eq. (56)] The condition (56) does not imply the separability claimed in Eq. (58). Let Q=g/h+3v_p and F=f+2gv_p+3hv_p^2. With the substitution w=Q e^{-\int F} \theta, which is the form consistent with Eq. (60), Eq. (31) becomes \theta'+(F-Q'/Q)\theta=hQ^2(\theta^2+\theta^3). To match Eq. (58) one needs F=Q'/Q+ShQ^2, hence f=Q'/Q+ShQ^2-2gv_p-3hv_p^2. The displayed condition (56) instead has (\ln Q)''+S(hQ^2)' in place of Q'/Q+ShQ^2, and these are not equivalent. For a concrete check, take h=1, g=y, v_p=0, S=1. The corrected condition gives f=y^2+1/y, and Eq. (44) is satisfied; the published condition (56) gives f=-1/y^2+2y, for which Eq. (44) fails. Therefore Theorem 3's hypothesis does not imply the claimed reduction, and Eq. (63) is unsupported as stated.
[II.A.3, Theorem 2] Theorem 2(a) states an integrability condition involving only f, g, and h, but the Abel equation (25) also contains k. The particular solution v_p defined by Eq. (47) is a solution of (25) only if Eq. (26) holds, i.e. k=dv_p/dy-fv_p-gv_p^2-hv_p^3. This consistency condition is absent from the theorem statement. As written, the theorem claims that any Abel equation whose coefficients satisfy (52) is integrable regardless of k, which is false. The theorem should either state the required compatibility condition on k or formulate the result for the Abel equation with this specific k.
[Theorem 2(b), Eq. (53)] Equation (53) is internally inconsistent with the derivation and with Eq. (55). In the E=1 case, the Chiellini substitution gives w=v-v_p=Q\theta with Q=g/h+3v_p, so v=v_p+Q\theta. Since v_p=-g/(3h)\pm\sqrt{g^2-3fh}/(3h), the correct formula is v=\pm[\sqrt{g^2-3fh}(3\theta+1)-g]/(3h), which matches Eq. (55). Equation (53) omits the v_p term and therefore gives w, not v. As printed, parts (b) and (c) of Theorem 2 cannot both be correct.
[II.B, Eq. (57)] The transformation displayed in Eq. (57) has e^{+\int F} in the prefactor, but the second equality in Eq. (60) and the preceding definition w=(v-v_p)e^{-\int F} require w=Q e^{-\int F}\theta. With the printed plus sign, the powers of E in Eq. (31) do not reconcile and the separable form (58) is not obtained even after the condition on f is corrected. This sign inconsistency should be fixed together with the corrected f-condition in Eq. (56).

minor comments (2)

[II.A.3, Eq. (52)] For the record, the integrated condition (52) is correct: writing B=g^2/(9h^2)-f/(3h), Eq. (51) gives B'/B^2=-18Sh, so 1/B=18S\int h\,dy+C, which is equivalent to (52) up to the naming of the constant. The reciprocal dependence on \int h\,dy is the correct outcome of the integration.
[III.A, Eq. (91)] The sign convention in Eq. (91) is ambiguous. For K=2 the separable integral gives \theta=1-1/(\int\sqrt{fk}\,dy+C), while Eq. (91) with the printed \mp symbols can be read as giving either -1/(R+C)+1 or -1/(R+C)-1. The authors should clarify the assignment of signs for K=2 and K=-2.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: integrability conditions are hypotheses and the solution formulas are derived by standard reductions; self-citations are contextual only.

full rationale

The paper's derivations are explicitly conditional. In Section II.A it states "We assume that a particular solution vp that satisfies Eq. (25) is known" (Eq. (26)), and all subsequent quadratures (Eqs. (34)-(38), (53)-(55), (61)-(63)) follow from the standard reduction v-vp = U, U = E w, followed by separation of variables once the Chiellini condition is imposed. The Chiellini conditions (44), (52), (78)-(79), and (84) are sufficient hypotheses on the coefficient functions, not outputs fitted to the solution formulas; the solution formulas are then derived algebraically from those hypotheses. Lemma 2 is proved in the paper rather than imported from the authors' earlier work, and the classical Chiellini lemma is cited to [28], an external source. Self-citations such as [29] and [43]-[45] appear only as contextual references in the introduction and are not load-bearing for the central claims. The skeptical observation that Eq. (56) appears to contain an algebraic error (the correct separation condition would be f = (ln Q)' + S h Q^2 - 2 g v_p - 3 h v_p^2, with Q = g/h + 3 v_p) is a correctness concern, not a circularity: even a corrected condition would be a hypothesis from which the separable equation (58) is derived. Hence no step reduces by construction to its own input, and the paper is self-contained in its derivation chain.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper uses classical results from the theory of Abel equations (Chiellini lemma, reduction using a particular solution) as axioms. No new entities are introduced. The constants S and P parametrize the integrability conditions and appear in the solution formulas.

free parameters (2)

S
Arbitrary constant in the Chiellini integrability conditions (Eqs. (39), (44), (66)-(67)); solutions depend on S.
P
Arbitrary constant in the generalized Chiellini Lemma (Eqs. (66)-(67), (72)-(77)).

assumptions (5)

standard math Chiellini integrability lemma for Abel equations
Quoted from Chiellini (1931) [28], used throughout as the core solution technique.
standard math Reduction of Abel equation using a known particular solution (Eqs. (15)-(18))
Standard result from Kamke [21]; used to transform the Abel equation after subtracting vp.
domain assumption Coefficients f,g,h,k are C∞ and nonzero on interval I; m,n>0, m≠n
Stated in Definition in Section II; restricts the class of equations.
domain assumption Existence of a particular solution vp of the Abel equation on the considered interval
Assumed in Theorems 1-3; without it the reduction to Eq. (27) is not available.
domain assumption y' is nonzero so that v = 1/y' is well-defined
Implicit in the transformation; solutions must be considered on intervals where y' does not vanish.

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Pith. "Pith review of On the integrability of the Abel and of the extended Li\'{e}nard equations." pith.science (2026). https://pith.science/paper/ZBNJ5DFK

@misc{pith2026190803730,
  author       = {Pith},
  title        = {Pith review of: On the integrability of the Abel and of the extended Li\'enard equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZBNJ5DFK}},
  note         = {Machine review of arXiv:1908.03730}
}

abstract

We present some exact integrability cases of the extended Li\'{e}nard equation $y^{\prime \prime }+f\left( y\right) \left(y^{\prime }\right)^{n}+k\left( y\right) \left(y^{\prime }\right)^{m}+g\left(y\right) y^{\prime }+h\left( y\right) =0$, with $n>0$ and $m>0$ arbitrary constants, while $f(y)$, $k(y)$, $g(y)$, and $h(y)$ are arbitrary functions. The solutions are obtained by transforming the equation Li\'{e}nard equation to an equivalent first kind first order Abel type equation given by $\frac{dv}{dy} =f\left( y\right) v^{3-n}+k\left( y\right) v^{3-m}+g\left( y\right) v^{2}+h\left( y\right) v^{3}$, with $v=1/y^{\prime }$. As a first step in our study we obtain three integrability cases of the extended quadratic-cubic Li\'{e}nard equation, corresponding to $n=2$ and $m=3$, by assuming that particular solutions of the associated Abel equation are known. Under this assumption the general solutions of the Abel and Li\'{e}nard equations with coefficients satisfying some differential conditions can be obtained in an exact closed form. With the use of the Chiellini integrability condition, we show that if a particular solution of the Abel equation is known, the general solution of the extended quadratic cubic Li\'{e}nard equation can be obtained by quadratures. The Chiellini integrability condition is extended to generalized Abel equations with $g(y)\equiv 0$ and $h(y)\equiv 0$, and arbitrary $n$ and $m$, thus allowing to obtain the general solution of the corresponding Li\'{e}nard equation. The application of the generalized Chiellini condition to the case of the reduced Riccati equation is also considered.

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and [41] for detailed discussions), one can associate to the Abe l equation the relative invariant S3 of weight 3, S3(x) ≡ s(x)p3(x) + 1 3 [ 2q2(x) 9 − r(x)q(x)p(x) + p(x)q′(x) − q(x)p′(x) ] . (8) S3 can be used to recursively generate an inﬁnite sequence of relative invariant...
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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

General solution of the Abel and extended Li´ enard equation f or vp = vp0 = −g(y)/ 3h(y) 6

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General solutions of Abel and extended Li´ enard equation for vp ⁄= −g(y)/ 3h(y) via the Chiellini integrability condition 6

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The general solution of the extended quadratic-cubic Li´ enar d for arbitrary particular solutions of the associated Abel equation 8 III

The case E(y) ≡ 1 7 B. The general solution of the extended quadratic-cubic Li´ enar d for arbitrary particular solutions of the associated Abel equation 8 III. General solution of the extended Li´ enard equation for arbitrary n, m, and g(y) = h(y) ≡ 0 9 A. The Chiellini integrability condition for the Riccati equation 11 IV. Concluding remarks 12 Acknowl...

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(30) becomes dw(y) dy = h (y) E2(y)w3(y), E (y) = e ∫ [f (y)−g2(y)/ 3h(y)]dy

General solution of the Abel and extended Li´ enard equati on for vp = vp0 = −g(y)/3h(y) If the particular solution of the Abel equation (25) is given by vp0(y) = − g(y) 3h(y) , (32) then Eq. (30) becomes dw(y) dy = h (y) E2(y)w3(y), E (y) = e ∫ [f (y)−g2(y)/ 3h(y)]dy. (33) Eq. (33) can be straightforwardly integrated, giving w(y) = ± 1 √ 2 √ C − ∫ h(y)e2...

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(32), vp ⁄= −g(y)/ 3h(y), a solution of Eq

General solutions of Abel and extended Li´ enard equation for vp ⁄= −g(y)/3h(y) via the Chiellini integrability condition If the particular solution of the generalized Abel equation associat ed to the extended Li´ enard equation is not given by Eq. (32), vp ⁄= −g(y)/ 3h(y), a solution of Eq. (31) can be found with the use of the Chiellini Lemm a, which ca...

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The case E(y) ≡ 1 If the particular solution of the Abel equation (25) satisﬁes the co ndition 3h(y)v2 p + 2g(y)vp + f (y) = 0 , (46) or vp(y) = − g(y) 3h(y) ± √ g2(y) 9h2(y) − f (y) 3h(y) , (47) then, without any loss of generality, one can choose E(y) ≡ 1. For this case the Chiellini integrability condition becomes d dy [ 1 vp(y) − vp0(y) ] = 9Sh(y) [vp...

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Li´ enard, ´Etude des oscillations entretenues, Revue g´ en´ erale de l’´ electricit´ e23, 901-912 (1928)

A. Li´ enard, ´Etude des oscillations entretenues, Revue g´ en´ erale de l’´ electricit´ e23, 901-912 (1928)

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