The integration of ordinary differential equations: factorization and transformations

Mathematics and Computers in Simulation 57 (2001) 175–195

The integration of ordinary differential equations: factorization and transformations Lev M. Berkovich Department of Algebra & Geometry, Samara State University, Samara, 443011, Russia

Abstract This paper is based on a uniform theory of factorization and transformation of nth (n ≥ 2) order ordinary differential equations (ODEs) that are used to constructively solve problems of integrability. This method of factorization of differential operators is developed not only in a base differential field, but also in its algebraic and transcendental extensions. For the first time, the method is extended to nonlinear equations. A new method of exact linearization is proposed that includes transformations used earlier. This method allows us to constructively study nonlinear and nonstationary problems in mathematical simulations with the help of a computer algebra system called REDUCE (as well as other systems). © 2001 Published by Elsevier Science B.V. on behalf of IMACS. Keywords: Factorization; Transformation; Differential resultant; Liouvillian solution; Exact linearization

1. Introduction In this paper, we present an analytical study of the problem of integrability of ordinary differential equations (ODEs). Two different approaches have been used for decades. One of these is related to variable changes and the other is applied to algebraic analogies. However, these substitutions have a heurestic nature, as a rule, and such a powerful tool as factorization is rarely used on differential equations (even linear ones) because it is inefficient. There were high expectations for the application of Lie groups and algebraic theory to differential equations (group analysis), and not in vain. The conceptual and uniform role of the theory is now universally recognized. It has proven to be especially fruitful in the conformity to fundamental equations of mechanics and physics, since invariance principles were used even when deriving the equations (see [1,2]); however, it does not allow the integrability problem “to close”. Much success is achieved when integrating nonlinear ODEs using Painlevé’s test; however, its entire capability is uncertain because the observable connection with integrability is insufficient at this time. On one hand, there is an urgent need to solve the integrability problem for differential equations because of a E-mail address: [email protected] (L.M. Berkovich). 0378-4754/01/$ – see front matter © 2001 Published by Elsevier Science B.V. on behalf of IMACS. PII: S 0 3 7 8 - 4 7 5 4 ( 0 1 ) 0 0 3 3 7 - 8

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necessity to obtain exact solutions for new mathematical models. On the other hand, there is a new awareness that integrability is an interdisciplinary field of knowledge with different aspects that stimulate the development of fundamental mathematical sciences, including algebra, geometry, and analysis. However, current achievements do not eliminate the necessity to solve such an inexhaustible problem as integrability. The author of this paper has concluded that the key to comprehending the integrability problem for ODEs is contained in the ideas of factorization and transformation. Moreover, it is necessary to combine both applications since the summarized results are most useful. As a result, new possibilities appear not only in the construction, on of new classes of integrable equations, but also in the explanation of many “vouders” of the integrability. The methods of group analysis and differential algebra that are discussed in this paper as well as factorization, autonomization, and exact linearization are currently being developed by the author. These tools are the basis for the construction of effective algorithms, and they are partially implemented by means of computer algebra systems, including REDUCE. The author is the first person to present the factorization method for differential operators in connection with the transformation theory of 1967 (see [3]). In this paper, factorization of differential operators is logically developed not only in a base differential field, but also in the algebraic and transcendental extensions (see also [4–7]). This is the first time the method is applied to nonlinear equations. The integrability theory is further developed, and a new method of exact linearization is proposed, which expands on earlier transformations. The audience for this paper will be specialists in differential equations, mathematical physics, group analysis, calculus, applied mathematics, mathematical simulations, and computer algebra, as well as theoretical and celestial mechanics. 2. Method of factorization for ordinary differential operators Our method of factorization of differential operators was developed after the concept of differential resultant was introduced. An example of this concept follows: A derivation D of a field F is map D: F → F satisfying D(a + b) = D(a) + D(b), D(ab) = D(a)b + aD(b) for all a, b ∈ F . An example is F = C(t), the rational functions in one variable over the complex numbers, with D = d/dx. (We consider the field F = C(x)). Definition 1.1. A differential field F0 is a pair (F, DF ) consisting of a field F and a designated derivation DF (usually denoted as D). The constants are members of the kernel of D; these form a subfield of F , i.e. number field k. (We consider k = C) (see [8]). A differential ring F0 [D] of differential operators is associated with F0 , but not commutative in general. Definition 1.2. The operator L is factorable (decomposable) in F0 , when it is a product of operators of lower orders with coefficients from F0 . The following definition is equivalent to the previous one. Definition 1.3. The equation Ly = 0 of order n is factorable in F0 if it has a nontrivial integral common with the equation My = 0 of order < n with coefficients from F0 . Otherwise, Ly = 0 is called nonfactorable in F0 .

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If the equations Ly = 0 and My = 0 have a common integral and the right, greatest common divisor RGCD(L, M) = L1 , then L = L2 L1 . In this case, the order of the equation Ly = 0 can be reduced, substituting L1 y = z, which leads to the equation L2 z = 0. Definition 1.4. The operator L isdecomposable to the prime factors (divisors) πi in F0 if there is
the factorization: L = 1k=l πksk , lk=1 mk sk = n, mi = ord πi , where πi does not allow any further factorization in F0 . Proposition 1.1. A selfadjoint differential operator creates the following factorization: L=

n   k=1

  1  2n + 1 − 2k 2n + 1 − 2k D+ α α . D− 2n − 1 2n − 1 k=n

This factorization can also be presented as a 2n-multiple iteration of a first-order operator, as follows:  2n       4n 2 α dx (D − α) , f (x) = exp α dx . fL = exp 2n − 1 2n − 1 Proposition 1.2. A antiselfadjoint differential operator allows the following factorization: L=

   1  n+1−k n+1−k α D α . D+ D− n n k=n

n   k=1

This factorization can also be written as the (2n + 1)-multiple iteration of a first-order operator:   2n+1      2n + 1 1 , f (x) = exp α dx (D − α) α dx . fL = exp n n Now, the operator equation and resultant matrices are considered (see [9]). Let L = ai D i , i = 0, n, ai ∈ F0 , n = ord L. The operator formally adjoint to L is denoted by L∗ and has the form   n k n k ∗ k k k L = (−1) D ak ≡ ak(s) D k−s . (−1) s k=0

k=0 s=0

The operators Lj ∈ F0 [D], j = 1, m, ord Lj = nj . Let us put n1 = maxj (nj ),nm = minj (nj ) and consider the equation (1.1) X1 L1 + X2 L2 + · · · + Xm Lm = 0, sj where Xj = k=0 xjk D k are the operators of orders n1 + nm − nj − 1 with undeterminated functional coefficients. Multiplying the operators in (1.1), we write it as n1 +n m −1 l=0

fl D l = 0,

(1.2)

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where the coefficients fl are linear forms of the unknowns xjk with the coefficients crk depending on aj and their derivatives, i.e. crk ∈ F0 . To determine xjk we have  to solve the system of linear algebraic equations consisting of n1 + nm equations with the following m j =1 (sj + 1) unknowns: fl (. . . , xjk , . . . ) = 0,

l = 0, n1 + nm − 1, j = 1, m, k = 0, sj .

(1.3)

Suppose the rank of (1.3) is r and equals the maximal order of an appropriate (basic) minor that is not equal to zero. Without loss of generality, the forms fl with indexes l = n1 + nm − r, . . . , n1 + nm − 1 can be regarded as linear and independent. Using the coefficients crk , system (1.3) is written in the form si m

xipi csi −pi ,li −pi (ai ) = 0,

l = 0, n1 + nm − 1.

(1.4)

i=1 pi =0

Definition 1.5. A matrix that coincides with the matrix in (1.4) up to transposition of the right resultant matrix R of the operators Li , i = 1, m. Thus, by definition, R = (M1 (a1 ), · · · , Mm (am ))T , where Mi (ai ) forms the following matrix:   csi n1 +nm −1 (ai ) csi n1 +nm −2 (ai ) ... csi 0 (ai )    ... csi −10 (ai )  0... csi −1n1 +nm −2 (ai )   Mi (ai ) =  . .. .. .. ..   . . . .   0...

...

c0,n1 (ai )

c00 (ai )

Definition 1.6. When the right resultant matrix of the operators L∗i , i = 1, m, is formally adjoint with Li , it is called the left resultant matrix R ∗ of the operator Li . For example, let us denote: r = rankR, r ∗ = rankR ∗ , d = ord RGCD(Lj ), and d ∗ = ord LGCD(Lj ) = ord RGCD(L∗j ). Theorem 1.1. The rank of the differential resultant matrix (DRM) is: n1 + nm = r + d, (n1 + nm = r ∗ + d ∗ ). Now, a differential analog of the Kronecker–Capelli theorem is given with respect to the following system: Li y = 0, i = 1, m,

(1.5)

and another system Li y = fi ,

i = 1, m.

(1.6)

The coefficients and left sides are sufficiently smooth functions in a real variable x, and at least one of the functions is fi = 0. Theorem 1.2. System (1.5) has the following nontrivial solution if r < n1 + nm .

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Theorem 1.3. System (1.6) is compatible if n¯ 1 + n¯ m − r¯ > n1 + nm − r. Here

 f i L¯ i = D − Li , fi



¯ n¯ j = ord L¯ j , n¯ 1 = max(n¯ j ), n¯ m = min(n¯ j ), r¯ = rank R. j

j

Automatic control problems led by Acad. Luzin [10] to investigate the compatibility of a system of ODEs. The idea of the DRM (see [9]) is explored in [11,12]. Definition 1.7. We say that # is the generalized Liouvillian (Eulerian) extension of the differential field F0 if there exists an ascending sequence of fields F0 ⊂ F1 ⊂ . . . ⊂ Fn = # such that one of the following conditions exists: 1. Fi = Fi−1 (α), where Fi−1 is a field of rational functions in α with coefficients from Fi−1 ; moreover, α ∈ Fi−1 (i.e. Fi is obtained from Fi−1 by joining the integral of an element of Fi−1 ). 2. Fi = Fi−1 (α), where α = 0 α /α ∈ Fi−1 (i.e. Fi is obtained from Fi−1 by joining the exponential integral of an element of Fi−1 ). 3. Fi = Fi−1 (α), where α is an algebraic element over Fi−1 (i.e. α satisfies an algebraic equation of degree n ≥ 2 with coefficients from Fi−1 ). 4. Fi = Fi−1 (y1 , y2 ), where y1 and y2 are linearly independent solutions of the second-order equation y + a1 y + a0 y = 0,

a1 , a0 ∈ Fi−1 .

(1.7)

If conditions 1 through 3 are fulfilled, then we have the Liouvillian extension #0 of the field F0 . If conditions 1 through 4 are fulfilled, then we have the generalized Liouvillian extension # of F0 . Thus, # contains #0 . Definition 1.8. If y1 , y2 , . . . , yn form a fundamental set of solutions (FSS) of the equation Ly ≡

n as y (s) = 0,

as ∈ F0 ,

(1.8)

s=0

then the differential field F0 (y1 , y2 , . . . , yn ), containing the algebraically closed field of constants with the characteristic zero is called the Picard–Vessiot Extension (PVE) for Eq. (1.8). Definition 1.9. Eq. (1.8) is integrated into a quadrature if PVE ⊂ #0 . Definition 1.10. Eq. (1.8) is integrated with (1.7) if PV ⊂ #. An algorithmic procedure exists for factorization of second-order ODEs in the algebraic extension of the field F0 . For example, the equation Ly ≡ y + a0 (x)y = 0,

a0 (x) ∈ F0

can admit the factorization Ly ≡ (D + α)(D − α)y = 0,

α = α(x),

(1.9)

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where α satisfies the Riccati equation α + α 2 + a(x) = 0. Proposition 1.4. Eq. (1.9) admits a factorization into a quadratic extension of the field F0 if the third-order antiselfadjoint differential operator L3 = D 3 + 4a(x)D + 2a (x) admits the factorization in F0 , as follows: L3 = (D 2 + pD + 2p + p 2 + 4a)(D − p),

p, a ∈ F0 .

Now we consider the factorization of operators in the transcendental Liouvillian extensions of the field F0 . 2.1. Factorization of Lame’s operator In Lame’s equation, y − (2℘ (x) + λ)y = 0,

λ = ℘ (α),

where ℘ (x) is the elliptic Weierstrass function. The factorization of Lame’s operator, L = D 2 − 2℘ (x) − ℘ (α), can be presented in two basic forms (see [13]), including: L = [D + ζ (x ± α) − ζ (x) ∓ ζ (α)][D − ζ (x ± α) + ζ (x) ± ζ (α)]. The factorization depends on the choice of the signs, where ζ (x) is the Weierstrass ζ -function. The generated cases of Lame’s equation lead to Schrödinger’s equations with rational, trigonometric and hyperbolic functions as shown below:       n(n + 1) n(n + 1) n(n + 1) y − λ+ y = 0, y − λ + y = 0, y = 0, y − λ + x2 cos2 x sin2 x     n(n + 1) n(n + 1) y − λ+ y = 0, y − λ + y = 0. cosh2 x sinh2 x Next, we consider the Lame–Halphen’s system of equations.  λ = ℘ (α), y − (2℘ (x) + λ)y = 0, y − 3℘ (x)y − ( 23 ℘ (x) + µ)y = 0,

µ = 21 ℘ (α).

The third-order Halphen operator LH = D 3 − 3℘ (x)D − ( 23 ℘ (x) + µ) admits the factorization LH = [D + ζ (x + α + β) − ζ (x) − ζ (α) − ζ (β)][D − ζ (x + α + β) + ζ (x + α) + ζ (β)] ×[D − ζ (x + α) + ζ (x) + ζ (α)],

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where ℘ (α), ℘ (β), and ℘ (γ ) are the roots of the characteristic equation ℘ 2 (x) − ℘ 2 (α) = 0,

℘ (α) + ℘ (β) + ℘ (γ ) = 0.

Compatibility with the Lame–Halphen’s system of equations is proved, and its eigenfunction ψ(x, λ, µ) = ˜ ψ(x, α) is found in the degenerate cases, where the coefficients are rational, trigonometrical, and hyperbolic functions as well as in the general case. 3. Kummer’s problem and liouvillian solutions of second-order linear equations In this section, the integrability theory is further developed. The author’s determination of Liouvillian solutions is based on research of the corresponding Kummer–Liouville transformation in explicit form (see [14,15]). The problem is also discussed in [16]. Let the equations y + a1 (x)y + a0 (x)y = 0,

a1 (x) ∈ C1 I, a0 (x) ∈ CI, I = {x|a < x < b},

(2.1)

b1 (t) ∈ C1 J, b0 (t) ∈ C(J ), J = {t|α < t < β},

(2.2)

and z¨ + b1 (t)˙z + b0 (t)z = 0,

where I and J are open (finite or not) intervals, and the Kummer–Liouville transformation (KLT) y = v(x)z,

dt = u(x)dx, v, u ∈ C2 I0 , uv = 0, ∀x ∈ I0 ⊂ I.

(2.3)

be given. (The latter is the most general point transformation, and it preserves the order and linearity of the equation (see [17])). Kummer’s problem is used to find the total set of local KLT (2.3) and then transform (2.1) to (2.2). Special attention is given to finding KLT. Global transformations of second-order linear ODEs are studied in [18]. Now, these conditions allow (2.1) to be reduced to (2.2). Theorem 2.1. Eq. (2.1) is reduced to (2.2) by transformation (2.3) if the following conditions exist (see [19]):     1 1 −1/2 v(x) = |u(x)| exp − a1 (x)(d)x + b1 (t)dt ; {t, x} + B0 (t)t 2 = A0 (x), t = u(x), 2 2 where {t, x} =

1 t 3 − 4 2 t



t t

2

is the Schwarz derivative; A0 (x) = a0 − 41 a12 − 21 a1 , and B0 (t) = b0 − 41 b12 − 21 b˙1

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are semi-invariants of Eqs. (2.1) and (2.2), respectively. The transformations of the depend variables y = µ(x)z, and z = ν(t)ξ, µ(x), ν(t) are sufficiently smooth functions;



v + a 1 v + a0 v − b 0 v

−3

  x  exp −2 a1 dx = 0, x0

b1 = 0,

  x   x −2  x v + a1 v + a0 v − b0 v −3 exp −2 a1 dx b1 (t (x))v −2 exp(− a1 dx)dx = 0, x0

x0

x0

where b1 = 0. Next, we discuss the reduction of equations with constant coefficients. Lemma 2.1. Eq. (2.1) is reduced to z¨ ± b1 z˙ + b0 z = 0,

b1 , b0 = constant

(2.2.1)

by the transformation (2.3), which admits the following factorizations: 1. In terms of first-order noncommutative operators:    v u v Ly ≡ D − − − r2 u D − − r1 u y = 0; v u v 2. In terms of first-order commutative operators:    1 v v 1 1 Ly ≡ D− − r2 D− − r1 y = 0, u2 u uv u uv where r1 and r2 are the roots of the characteristic equation r 2 ± b1 r + b0 = 0. Theorem 2.2. Eq. (2.1) is reduced to (2.2.1) by KLT; moreover, 1. Eq. (2.1) admits a one-parameter Lie group with the generator X = ξ(x, y)

∂ ∂ 1 ∂ v ∂ + η(x, y) ≡ + y ; ∂x ∂y u ∂x uv ∂y

2. u(x) satisfies the equation   1 u 3 u 2 1 2 − − δu = A0 (x), 2 u 4 u 4

δ = b12 − 4b0 ;

(2.4)

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3. the factor v(x) and kernel u(x) of KLT are connected to the following relations:     1 1 −1/2 v(x) = |u(x)| a1 (x)dx ± b1 u dx , v + a1 v + a0 v − b0 u2 v = 0; exp − 2 2 (2.5) and 4. v satisfies one of the following nonlinear equations:   x  v + a1 v + a0 v − b0 v −3 exp −2 a1 dx = 0, x0

b1 = 0,

(2.6)

or   x   x   x  −2 v +a1 v +a0 v − b0 v −3 exp −2 a1 dx b1 v −2 exp − a1 dx dx = 0, x0

x0

x0

b1 = 0;

 5. the function R(x) = exp(− a1 dx)u−1 is a resolvent of Eq. (2.1) and satisfies the equation R + 3a1 R + (4a0 + a1 + 2a12 )R + (2a0 + 4a0 a1 )R = 0. Thus, Kummer’s problem is related to nonlinear equations and a principle called nonlinear superposition hold. Definition 2.1. The principle of nonlinear superposition applies to the ODE F (x, y, y , . . . , y (n) ) = 0 if the general solution is represented as a nonlinear function in one of the following ways: (a) particular solutions of a nonlinear equation; (b) arbitrary constants; or (c) particular solutions of an associated linear equation. Now we consider Ermakov’s equation (see [20]) v + a0 (x)v − b0 v −3 = 0. This result was also obtained by [21] at a later date. In the present work, it follows from (2.6) that a1 = 0. Proposition 2.2. Ermakov’s equation has the general solution  v(x) = Ay22 + By2 y1 + Cy21 , δ = B 2 − 4AC = −4b0 , where y1 , y2 = y1



y1−2 dx

forms a FSS of Eq. (1.9). In the Kummer–Schwarz equation of the second-order (2.4), (KS-2) is integrated.

(2.7)

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Theorem 2.3. Eq. (2.4) has the following general solution, depending on δ: u(1) (x) = λ(x)(α1 y2 + β1 y1 )−1 (α2 y2 + β2 y1 )−1 , δ1 = (α1 β2 − α2 β1 )2 > 0; u(2) (x) = λ(x)(Ay22 + By1 y2 + Cy21 )−1 , δ2 = B 2 − 4AC < 0; u(3) (x) = λ(x)(αy2 + βy1 )−2 , δ3 = 0. The following special cases also need to be considered: u(4) (x) = λ(x)(αy2 + βy1 )−1 yi−1 ,

i = 1, 2; δ4 = α 2 ;

u(5) (x) = λ(x)yi−2 , δ5 = 0,

where 

λ(x) = e− a1 dx ,  y2 = y1 λ(x)y1−2 dx, i.e. y1 and y2 form a FSS of Eq. (2.1).

(2.8)

Now point symmetries of second-order linear equations are defined. See [22] for a comparable discussion of point symmetries. Definition 2.2. A set G of the differentiable and reversible point transformations Ta : x1 = f (x, y; a), y1 = ϕ(x, y; a),

(2.9)

where a is the parameter, f and ϕ are expanded in Taylor series in neighborhood of a = 0, form a local one-parameter Lie group, if the following conditions hold: (1) T0 = Id ∈ G,

(2) Tb Ta = Ta+b ∈ G,

(3) Ta−1 = T−a ∈ G,

(4)(Ta Tb )Tc = Ta (Tb Tc ) ∈ G

for sufficiently small a, b, and c. Definition 2.3. The equation F (x, y, y , . . . , y (n) ) = 0 admits the one-parameter Lie symmetry group G given in (2.9), if the equation is invariant with respect to the extended group G(n) . The Lie algebra of symmetries for Eq. (2.1) is 8-dimensional, isomorphic to sl(3, R), and has generators (infinitesimal operators and vector fields) of the form  1 ∂ v ∂ ∂ ∂ X1 = + y ; X2 = v = y1 ; X3 = X1 u dx; u ∂x uv ∂y ∂y ∂y  ∂ y y ∂ X4 = X2 u dx = y2 ; X5 = X1 , X6 = X2 = y , ∂y v v ∂y  2      c  y 2 y y X7 = u dx X1 + u dx X2 , X8 = X2 , X1 + v v v where u(x) and v(x) satisfy the equations, as follows:      1 u 3 u 2 1 −1/2 = A0 (x), v(x) = |u(x)| exp − − a1 (x)dx . 2 u 4 u 2

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Now the basic for the KLT Eq. (2.5), (B-equation), is integrated. This linear equation is associated with (2.1), and written as follows: v + a1 v + a0 v − b0 λ2 (x)(ay22 + by2 y1 + cy21 )−2 v = 0,

(2.5.1)

where y1 and y2 form a FSS of Eq. (2.1), and λ(x) satisfies (2.8). B-equation (2.5.1) has the general solution √

√

(1) v(1,2) = (α1 y2 + β1 y1 )1/2±b1 /(2 δ1 ) (α2 y2 + β2 y1 )1/2∓b1 /(2 δ1 ) , δ1 > 0;    b1 2Ay2 + By1 (2) 2 2 v(1,2) = Ay2 + By2 y1 + Cy1 exp ± √ , δ2 < 0; arctan √ −δ 2 −δ 2 y1   b1 y1 (3) v(1,2) = (αy2 + βy1 ) exp ∓ , δ3 = 0; 2α(αy2 + βy1 ) 1/2∓b /(2α)

(4) 1 = (αy2 + βy1 )1/2±b1 /(2α) yi v(1,2)   b1 y2 (5) v(1,2) = yi exp ± , δ5 = 0. 2 y1

, δ4 = α 2 > 0, i = 1, 2;

If a1 = 0 then B-equation (2.5) becomes v + (a0 (x) − b0 u2 (x))v = 0.

(2.10)

In addition, if b1 = 0, then B-equation (2.10) is Ermakov’s Eq. (2.7). 4. Related second-order linear ODE (Kummer–Liouville’s and Euler–Imshenetskii–Darboux’s transformations) In this section, a new algorithmic procedure is proposed to construct a sequence of linear second-order ODEs that are integrable in terms of a given equation. Next, a technique is described “to reproduce” equations of the form (1.9), denoted by (a0 ), using KLT indirectly. Proposition 3.1. Equation (a0 ) induces the sequence of equations (ak ) as follows (see [23]): yk + ak yk = 0,

(3.1)

where ak = a0 −

k

b0s u2s ,

b0s = constant = 0,

ak = ak−1 − b0k u2k ,

s=1

and the function us (x) satisfies the following sequence of (KS-2) equations:   1 u s 3 u s 2 1 2 − − δs us = as−1 . 2 us 4 us 4 2 In addition, δs = b1s − 4b0s are the discriminants of the characteristic equation:

rs2 ± b1s rs + b0s = 0.

(3.2)

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Then, linearly independent solutions yk(1,2) are      1 −1/2 exp ± yk(1,2) = |uk | b1k uk dx , 2  −1/2 yk2 = |uk | uk dx, b1k = 0.

b1k = 0,

yk1 = |uk |−1/2 ,

The sequence of equations (ak ) can be presented as   k yk + a0 − b0s (αs1 y2s−1 + βs1 y1s−1 )−2 (αs2 y2s−1 + βs2 y1s−1 )−2 yk = 0,

(3.3)

s=1

where y1s and y2s form a FSS of the equation ys + as ys = 0,

(αs1 βs2 − αs2 βs1 )2 = δs .

The sequence of (KS-2) equations in (3.2) is analogous to the following sequence of Ermakov’s equations: vs + as−1 vs − b0s vs−3 = 0. The following sequence of resolvent equations of the third-order (for the proper second-order linear equations (as )) Rs + 4as Rs + 2as Rs = 0 and a sequence of (KS-3) equations {ts , x} + as ts 2 = as−1 are analogous. The base sequence of related equations are induced by the equation y = 0 (i.e by (3.3) with a0 = 0). Example 3.1. Liouville’s equation follows: y + d(ax2 + bx + c)−2 y = 0. This equation corresponds to the base equation v − b0 u2 v = 0,

d = −b0 ,

u(x) = (ax2 + bx + c)−1 .

The following two examples, which arose from research of a radial Schrödinger equation for a particle in the central force field (see [24]), belong to the (0)-sequence. Example 3.2.   m(m + 1) 1 y − + 4 y = 0, x2 T Example 3.3.   1 1 + y = 0, y + 4x 2 x 2 S 4

T = αx −m + βx m+1 ,

s = α log x + β.

The following two equations also belong to the (0)-sequence:

1 m = − . 2

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Example 3.4. y + m2 y + d(a sin2 mx + b sin mx cos mx + c cos2 mx)−2 y = 0. Example 3.5. y − m2 y + d(a sinh2 mx + b sinh mx cosh mx + c cosh2 mx)−2 y = 0. Examples 3.4 and 3.5 are contained in the class of Ince–Schrödinger’s equations. Example 3.6. The equations depend on an arbitrary function:



−4

−2

−2

y − [f (x) + f (x) + b01 ϕ (α1 Φ + β1 ) (α2 Φ + β2 ) ]y = 0, where Φ=

2

   ϕ = exp − f dx ,

 ϕ 2 dx.

Now, related linear ODEs, (a0 ) and (ak ), are studied. These equations are connected to each other by the following convertible Euler–Imshenetskii–Darboux’s transformation (EIDT), (see [25–27]): z = β(x)y − α(x)y,

β(x), α(x) ∈ C2 I,

β = 0,

α β − αβ + α 2 + a0 β 2 = C = 0.

4.1. Operator identities Let us introduce the differential operators A = D 2 + a0 ,

Ak = D 2 + ak ,

Lk =

k 

(D − αs−1 ),

(3.4)

s=1

where ak can be written as:  k k  k y˜s−1 ak = a0 + 2 (ln y˜s−1 ) = a0 + 2 = a0 + 2 αs−1 , y ˜ s−1 s=1 s=1 s=1 y˜s−1 is the eigenfunction of the equation ys−1 + (as−1 − λ)ys−1 = 0,

which corresponds to the eigenvalue λ = λs−1 , and αs−1 satisfies Riccati’s equation 2 αs−1 + αs−1 + as−1 = λs−1 .

Theorem 3.1. Solutions of the operator equations Xk A = Ak Xk ,

k = 1, 2, . . .

(3.5)

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are the operators Xk = Lk , and solutions of the equations Ak y = 0 are the functions yk (x) = Lk y0 (x), where y0 (x) is a solution of the equation Ay = 0, and the operators A, Ak , and Lk are determined by (3.4) and (3.5). This theorem reinforces the classic result concerning “reproduction” of linear second-order ODEs (LODE-2) using EIDT. Now, Euler’s problem for complete LODE-2 (3.1) is considered. Moreover, a connection between KLT and EIDT is established. Theorem 3.2. Eq. (3.1) is transformed to itself, i.e. reduced to the form z + a1 (x)z + a0 (x)z = 0,

a1 (x) ∈ C1 (I),

a0 (x) ∈ C(I),

by means of EIDT (3.4), where α β − αβ + α 2 + a0 β 2 + a1 αβ = C = 0, if β(x) = u−1 (x),

α(x) = v v −1 u−1 ,

where u(x) and v(x) are the kernel and factor of KLT that transforms (3.1) to an equation with constant coefficients.

5. A new method for exact linearization of ODEs The uniform theory of factorization and transformation is the basis for constructively solving equations of the order n (n ≥ 2) equivalence. This method is used to find the general form of nonlinear nth (n ≥ 2) order equations while admitting exact linearization by means of a nonlocal transformation of dependent and independent variables. This is the first time these variables have been used to factorize nonlinear ODEs. Theorem 4.1. The equation(see [7,28]) F ≡ y (n) − f (x, y, y , . . . , y (n−1) ) = 0

(4.1)

is reduced to the linear autonomous form z(n) (t) + bn−1 z(n−1) (t) + . . . + b1 z (t) + b0 z(t) = 0,

bk = constant,

(4.2)

by means of the transformation y = v(x, y)z,

dt = u1 (x, y)dx + u2 (x, y)dy,

(4.3)

where v, u1 , and u2 are sufficiently smooth functions in a domain Γ (x, y), within which v(u1 +u2 y ) = 0, iff (4.1) admits the noncommutative factorization F ∼

1   k=n

 v x + vy y D(u1 + u2 y ) D− − (k − 1) − rk (u1 + u2 y ) y = 0, v u1 + u 2 y

L.M. Berkovich / Mathematics and Computers in Simulation 57 (2001) 175–195

or the commutative one  n   vx + vy y 1 − r D − F ∼ k y = 0, ) v(u u + u y + u y 1 1 2 2 k=1

D=

d , dx

vx =

∂v , ∂x

189

vy =

∂v , ∂y

where rk is the root of the characteristic equation r n + bn−1 r n−1 + . . . + b1 r + b0 = 0. An essential component of transformation (4.3) is: KLT (2.3), the general point linearization   ∂(t, z) t = f (x, y), z = ϕ(x, y), J = tx zy − ty zx = 0, ∂(x, y)

(4.4)

which corresponds to (4.3) for u1y = u2x , the point linearization t = f (x),

z = ϕ(x, y);

(4.5)

which preserves fibering and nonlocal linearization of nonlinear autonomous equations y = v(y)z,

dt = u(y)dx,

u(y(x))v(y(x)) = 0,

∀x ∈ I = x|a ≤ x ≤ |;

(4.6)

the linearization y = v(x, y)z, dt = u(x, y)dx, connected with arbitrary point Lie symmetry; and finally, general nonlocal linearization (4.3). Theorem 4.2. Eq. (4.1) is reduced to (4.2) when we combine the transformation KLT with (4.6), as follows (see [6]):     y y y = v1 (x)v2 (x) z, dt = u1 (x)u2 (x) dx v1 v1 if the noncommutative factorization 1   k=n

 v1 u 1 v2∗ u∗2 D− − (k − 1) − Y − (k − 1) Y − rk u1 u2 y = 0, v1 u1 v2 u2

or the commutative one  n   v1 v2 + v1 v2∗ Y 1 D− − rk y = 0, u1 u2 v1 v2 u 1 u 2 k=1 holds; besides, the diagram f

A ↓ϕ

→

C

→

ψ

B ↓g D

Y =

y , v1

( ) =

d , dx

(4.7)

(∗) =

d ; dY

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is commutative because g ◦ f = ψ ◦ ϕ. Here, A denotes the set of Eqs. (4.1) and (4.3), B denotes the set of nonlinear autonomous equations having the factorized form  1   u∗2 dY v2∗ dY d − (k − 1) − rk u2 Y = 0, Ds = ; Ds − v2 ds u2 ds ds k=n C denotes a set of linear nonautonomous (and reducible) equations  1   dV1 /dq d dU1 /dq Dq − , − (k − 1) − rk U1 p = 0, Dq = V U dq 1 1 k=n

v1 (x) = V1 (q(x)),

u1 (x) = U1 (q(x)), and D denotes the set of linear equations with constant coefficients (4.2). Transformations f, g, ϕ, and ψ are respectively: f : y = v1 (x)Y, ds = u1 (x)dx; g : Y = v2 (Y )z, dt = u2 (Y )ds;     y y ϕ : y = v2 p, dq = u2 dx; ψ : p = v1 (x)z, dt = u1 (x)dq. v1 v1 In the following th. 4.3, the proposed method is used for second and third-order equations, and we discuss the linearization of second-order autonomous equations (see [29]). Linearization by means of a transformation of the unknown function is used in [30], and the independent variable is used in [31,32]. Additional examples are given in [33]. Theorem 4.3. The equation d , dx is linearized by means of transformation (4.6), i.e. reduced to y + f (y)y 2 + b1 ϕ(y)y + ψ(y) = 0,

( ) =

d , b1 , b0 , c = constant, dt if it can be represented in one of the following forms:        c 2 = 0, y + fy + b1 ϕy + ϕ exp − f (y)dy) b0 ϕ exp( f (y)dy dy + β   2a ϕ∗ b0 c y − + y 2 + b1 ϕy + ϕ 2 y(ay + b) + ϕ 2 (ay + b)2 = 0, ay + b ϕ b b z¨ + b1 z˙ + b0 z + c = 0,

()˙ =

(4.8)

((∗) = d/dy, and β = constant is normalizing factor) which are reduced to (4.8) by the transformations, respectively    z = β ϕ exp f dy dy, dt = ϕ(y)dx; and z=

y , ay + b

dt = ϕ(y)dx.

L.M. Berkovich / Mathematics and Computers in Simulation 57 (2001) 175–195

191

Some classes of second-order dynamical systems can be linearized. Particularly, the equation y + f (y)y 2 ± a 2 ψ(y) = 0 is linearized by the transformation      z=

2

ψ exp 2



f dy dy,

dt = ψ(exp



fdy z−1 dx,

to the form of harmonic oscillator z¨ ± a 2 z = 0, and the first integrals and one-parameter families of solutions are respectively:          2 2 y = a C ∓ 2 ψ exp 2 f dy dy exp −2 (f dy ,    exp( f dy)dy = ± ∓a 2 x + c. z Now we discuss linearization of a third-order autonomous equation of the form: y + f5 (y)y y + f4 (y)y + f3 (y)y 3 + f2 (y)y 2 + f1 (y)y + f0 (y) = 0. Let Eq. (4.9) be reduced to ... z + b2 z¨ + b1 z˙ + b0 z + c = 0,

b2 , b1 , b0 , c = constant.

(4.9)

(4.10)

by transformation (4.6). Theorem 4.4. Eq. (4.9) is linearized by the transformation (4.6), if it is represented as  ∗∗    1 ϕ ϕ∗ ϕ ∗2 1 ϕ∗ 3 2 ∗ y + f (y)y y + 3 −5 2 −f + f + 3f y + b2 ϕy + b2 ϕ f + y 2 9 ϕ ϕ ϕ 3 ϕ          c 1 1 2 5/3 4/3 +b1 ϕ y + ϕ f dy dy + exp − f dy = 0. (4.11) b0 ϕ exp 3 β 3 which is reduced to (4.10) by the transformation     1 4/3 f dy dy, dt = ϕ(y)dx. z = ϕ exp 3 Example 4.1. The discovery of an invariant solution of the sin-Gordon equation (see [34], pp. 117–119). uxt = sin u has led to the ODE 1 y + y 3 = 0. 2

(4.12)

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L.M. Berkovich / Mathematics and Computers in Simulation 57 (2001) 175–195

... In this special case, (4.11) is linearized to z = 0 by the substitution   3 z = exp(2iy), dt = exp iy dx. 2 The general solution of (4.12) in parametric form is  1 x = (c1 + c2 t + c3 t 2 )−3/4 dt, y = − i ln (c1 + c2 t + c3 t 2 ). 2 Now the classic Euler–Poinsot problem of solid rotation around a fixed point is considered from a new perspective. It is already known that this problem can be characterized by the system Ap˙ − (B − C)qr = 0, C r˙ − (A − B)pq = 0,

B q˙ − (C − A)rp = 0, A, B, C = constant,

(4.13)

which separates the variables. After we eliminate in (4.13) the variables q and r, and change the notation to (.) → ( )

t → x, p → y,

the result is the third-order equation y −

1 4(A − B)(C − A) 2 yy − y y = 0. y BC

(4.14)

In the latter case, (4.14) is linearized to ... z + b˙z = 0,

b=−

4(A − B)(C − A) >0 BC

by the transformation z = y2,

dt = ydx.

(4.15)

Next, we discuss point linearization of autonomous and nonautonomous second-order equations, which are compared to nonlocal linearization. Even [35,36] has circumscribed this class of ODEs: y = F (x, y · y ),

(4.16)

is linearizable by the point substitution (4.4) and reducible to z¨ = 0.

(4.17)

Theorem 4.5. The most general second-order equation that is reducible to the linear form(4.8) by point transformation (4.4) is (fx ϕy − ϕx fy )y + [(fy ϕyy ) − ϕy fyy ) + b1 ϕy fy2 + (b0 ϕ + c)fy3 ]y 3 + [fx ϕyy − ϕx fyy +2(fy ϕxy − ϕy fxy ) + b1 (ϕx fy2 + 2fx fy ϕy ) + 3(b0 ϕ + c)fx fy2 ]y 2 +[fy ϕxx − ϕy fxx + 2(fx ϕxy − ϕx fxy ) + b1 (2fx fy ϕx + fx2 ϕy )

+3(b0 ϕ + c)fx2 fy ]y + (fx ϕxx − ϕx fxx ) + b1 ϕx fx2 + (b0 ϕ + c)fx3 = 0.

L.M. Berkovich / Mathematics and Computers in Simulation 57 (2001) 175–195

193

Lemma 4.1. Eq. (4.16) is reduced to (4.17) by the transformation (4.5) if the following conditions hold:     ∂f2 ∂f1 1  f2 ∂y 2 − f2 ∂y ∂f1 1 2 F = f2 (x, y)y +f1 (x, y)y +f0 (x, y), e =2 , f0 = e − f ∂y. ∂y ∂x 2 ∂x 2 1 This statement reinforces the result of Hsu and Kamran [37]. Now we consider the integrability of the following equations: b y + ayy + y 3 = 0, 2

a = constant,

b = constant,

(4.18)

These equations are related to analytical theory of ODEs, integrals of nonlinear second-order ODEs (see [38]), and several problems of theoretical and mathematical physics (see [39]). The Chebyshev criterium of integration of differential binomials (see [40], p. 66.) is applied to the integration of Eq. (4.18) in closed form along with the exact linearization technique. Note that Eq. (4.18) for all a and b is linearized to z¨ + a z˙ + bz = 0 by nonpoint transformation (4.15) it while the linearization by point transformation is possible only for b=

2 2 a , 9

and it is reduced to (4.17) by the substitution 1 1 t = ax − , 3 y

1 x z = ax2 − . 6 y

The equation y − γ y 2 − βyy − αy + y − δy 2 = 0,

(4.19)

is encountered in qualitative theory of ODEs, e.g. in auto-oscillation theory (see [41]). In conclusion, the application of exact linearization and factorization can be used to find first integrals and one-parameter families of integral curves explicitly. Theorem 4.6. The Eq. (4.19) has first integrals and one-parameter families of solutions in the following cases: 1. For γ = 0: 1.1. γ = −β/α, δ = γ , δ = 0, 1.2. α = β = 0; γ = −δ. 2. For γ = 0, β = 0: 2.1. 4δ 2 + β 2 + 2αβδ = 0; 2.2. δ 2 + β 2 + αβδ = 0; 2.3. β = constant = 0. √ 3. For γ = 0, β = 0, α = ±5/ 6.

α = 0; α = 0,

δ = −γ ;

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