Electronic Journal of Qualitative Theory of Differential Equations 2007, No. 20, 1-12; http://www.math.u-szeged.hu/ejqtde/

On Rectifiable Oscillation of Euler Type

Second Order Linear Differential Equations

James S.W. Wong Institute of Mathematical Research, Department of Mathematics, The University of Hong Kong E-mail address: jsww@chinneyhonkwok.com

ABSTRACT We study the oscillatory behavior of solutions of the second order linear differential

equation of Euler type: (E) y′′ + λx^{−α}y = 0, x ∈ (0, 1], where λ > 0 and α > 2. Theorem (a) For 2 ≤ α < 4, all solution curves of (E) have finite arc length; (b) For α ≥ 4, all solution curves of (E) have infinite arc length. This answers an open problem posed by M. Pasic [8]

§1. In a recent paper [8], M. Pasic introduced the concept of “ rectifiable oscillation” in the study of oscillatory behavior of solutions of the second order linear differential equation of Euler type in the finite interval (0, 1]:

(1) y′′ + λx^{−α}y = 0, x ∈ (0, 1].

In case of the Euler’s equations, i.e. α = 2, it is well known that equation (1) is oscillatory when λ > \frac{1}{4} and nonoscillatory if λ ≤ \frac{1}{4}. For α > 2, it follows from the Sturm’s Comparison Theorem that all solutions of (1) are oscillatory as x → 0.

For a real function y(x) defined on the closed interval \overline{I} = [0, 1], we denote its graph by G(y) = {(t, y(t)) : 0 ≤ t ≤ 1} as a subset in ℝ^{2}. G(y) is said to be a rectifiable curve in ℝ^{2} if its arc length L_{G}(y) is finite where L_{G}(y) is defined by

L_{G}(y) = sup {∑_{i=1}^{m} ∥(t_{i}, y(t_{i})) − (t_{i−1}, y(t_{i−1}))∥_{2}} ,

where supremum is taken over all partitions: 0 = t_{0} < t_{1} < ⋯ < t_{m} = 1 of the unit interval [0, 1] (See Apostol [1], p. 175). Here ∥ · ∥_{2} denotes the Euclidean norm in ℝ^{2}.

An oscillatory function y(x) on I = (0, 1] is said to be rectifiable (resp. unrectifiable) if its graph G(y) is rectifiable (resp. unrectifiable). In other words, it is rectifiable if it has finite arc length on I and unrectifiable otherwise. Equation (1) is said to be rectifiable (resp. unrectifiable) oscillatory on I if all its nontrivial solutions are rectifiable (resp. unrectifiable).

In the familiar case of the Euler’s equation, i.e. equation (1) when α = 2 and λ > \frac{1}{4}, we can write down the explicit form of its general solutions

(2)  y(x) = c_{1}\sqrt{x} cos(ρ log x) + c_{2}\sqrt{x} sin(ρ log x),

where ρ^{2} = λ − \frac{1}{4}.

Consider a typical section of the solution curve y(x) of (1) between its two zeros a_{k+1} and a_{k} where a_{k+1} < a_{k} and a_{k} → 0 as k → ∞. It is clear from the geometry that its arc length is bounded above by 2|y(s_{k})| + a_{k} − a_{k+1} where y′(s_{k}) = 0 and a_{k+1} < s_{k} < a_{k}. In the special case (2) of the Euler equation, we can simply estimate sin(ρ log x) as the case of cos(ρ log x) is similar. For x ∈ (0, 1], log x is negative, so zeros of sin(ρ log x) are given by a_{k} = exp(−kπ/ρ) and maxima of sin(ρ log x) occurs at s_{k} = exp(−kπ/2ρ). The curve sin(ρ log x) over interval (a_{k+1}, a_{k}) is bounded by 2|\sqrt{s_{k}}| and a_{k} − a_{k+1}, hence

L_{G}(y) ≤ 2 ∑_{k=0}^{∞} exp(−kπ/4ρ) + 1. < ∞,

so equation (1) when α = 2, λ > \frac{1}{4} is rectifiable oscillatory.

When α = 4, equation (1) can also be solved in explicit form, namely, solutions are of the form

(3)  y(x) = c_{1}x cos(\sqrt{λ}/x) + c_{2}x sin(\sqrt{λ}/x).

The typical section of the solution curve y(x) between two zeros a_{k+1} and a_{k} with its extrema at s_{k}, a_{k+1} < s_{k} < a_{k}, has arc length exceeding 2|y(s_{k})|. Let us again consider only the case y(x) = x cos(\sqrt{λ}/x) which has zeros at a_{k} = 2\sqrt{λ}/(2k + 1)π and at its extrema s_{k}, we have

EJQTDE, 2007 No. 20, p. 2

|y(s_{k})| ≥ |y(\sqrt{λ}/kπ)| = \sqrt{π}/kπ. Clearly ∑_{k=0}^{∞} |y(s_{k})| > \frac{\sqrt{λ}}{π} ∑_{k=0}^{∞} \frac{1}{k} = ∞, so the length of of the

solution curve x cos(\sqrt{λ}/x) is infinite hence the oscillatory solution given in (3) is unrectifiable. This is interesting because the solution curves given in (3) exhibits a phenomenon whereby a one dimensional curve can have infinite length within a bounded region in ℝ^{2}, like the Koch’s snowflake in fractal geometry, see Devancy [5] p. 182.

Pasic [8] introduced the following two boundary layer conditions:

(I) there exist c > 0 and d ∈ (0, a] depending on y(x) such that |y′(x)| ≤ cx^{−α/4} for all

x ∈ (0, d).

(II) there exist c > 0 and d ∈ (0, a] depending on y(x) that |y(x)| ≤ cx^{α/4} for all

x ∈ (0, d),

and proved

Theorem ℙ. For solutions of (1) satisfying boundary layer condition (I) (or (II)), we have for λ > 0,

(a) if 4 > α > 2; then equation (1) is rectifiable oscillatory:

(b) if α ≥ 4, then equation (1) is unrectifiable oscillatory provided that it also admits the existence of two linearly independent solutions satisfying boundary layer condition (I) (or (II)).

Pasic [8] posed as an open problem whether the additional assumptions of boundary layer conditions (I) and (II) can be removed for other values of α, α ≠ 2 and 4. The purpose of this paper is to show that both of these boundary layer conditions are superfluorous in the case of equation (1). We can therefore improve Theorem P to read

Theorem 1. (a) Equation (1) is rectifiable oscillatory for 2 < α < 4, and (b) Equation (1) is unrectifiable oscillatory for α ≥ 4.

§2. In this section, we shall give the proof of our main theorem. In view of Theorem ℙ, it suffices to prove that solutions of (1) satisfy either one of the boundary layer conditions (I) and (II). This can be accomplished by the use of Louiville transformation and an application

EJQTDE, 2007 No. 20, p. 3 of an asymptotic integration theorem in the elliptic case.

let y(x) = x^{α/4}z(x), then z(x) satisfies, by equation (1), the following second order linear differential equation,

(4)  (x^{α/2}z′)′ + \{λx^{−α/2} + \frac{α}{4} (\frac{α}{4} − 1) x^{−2+α/2}\} z = 0, x ∈ (0, 1].

We make the Louiville transformation for α > 2

(5)  s = (\frac{α}{2} − 1)^{−1} x^{1−α/2} ẑ(s) = z(x),

which transform the behavior of x near 0 to s near infinity. Using (5), equation (4) becomes

(6)  \frac{d^{2}ẑ}{ds^{2}} + [λ + \frac{α}{4} (\frac{α}{4} − 1) (\frac{α}{2} − 1)^{−2} \frac{1}{s^{2}} ] ẑ = 0, s ≥ 1.

Solutions of (6) have the form

(7)  ẑ(s) = (c_{1} + o(1)) sin \sqrt{λ}s + (c_{2} + o(1)) cos \sqrt{λ}s

see Hartman [6; Chapter XI Theorem 8.1, p.370]. In particular, \frac{dẑ}{ds}(s) has the similar form (7) because the unperturbed equation

(8)  \frac{d^{2}û}{ds^{2}} + λû = 0, s ≥ 1

has sin \sqrt{λ}s and cos \sqrt{λ}s as fundamental solutions. Since ẑ(s) is bounded by (7) so by (5) and the transformation y(x) = x^{α/4}z(x), we deduce that y(x) = 0 (x^{λ/4}) satisfying boundary layer condition (II). To see that boundary layer condition (I) is also satisfied, we note that

(9)  y′(x) = x^{α/4}z′(x) + \frac{α}{4}x^{α/4−1}z(x)

and

(10)  z′(x) = −x^{−α/2} \frac{dz}{ds} (s) = 0 (x^{−α/2}) .

From (9) and (10), we obtain y′(x) = 0 (x^{−α/4}) since for α ≥ 2, \frac{α}{4} − 1 ≥ −\frac{1}{2} ≥ −\frac{α}{4} and 0 (x^{α/4−1}) = 0 (x^{−α/4}) as x → 0. This completes the proof that all solutions of (1) when α > 2 satisfy both the boundary layer conditions (I) and (II).

EJQTDE, 2007 No. 20, p. 4

We can now apply Theorem ℙ to complete the proof of our main Theorem. On the other hand, by using the transformed equation (4), we can also give a much simpler proof somewhat different from that gives by Pasic [8].

We first consider the unperturbed equation (8)

(11)  (x^{α/2}u′)′ + λx^{−α/2}u = 0, û(s) = u(x)

which has its general solution given by

u(x) = c_{1} sin (\sqrt{λ}/σx^{σ}) + c_{2} cos (\sqrt{λ}/σx^{σ}) , σ = (\frac{α}{2} − 1) > 0,

Let \{x_{n}\} be the sequence of consecutive zeros of u(x) = sin (\sqrt{λ}/σx^{σ}), i.e.

(12)  x_{n} = (\frac{\sqrt{λ}}{σnπ} )^{1/σ}

We denote q(x) = \{λx^{−α/2} + \frac{ασ^{−2}}{4} (\frac{α}{4} − 1) x^{−2+α/2}\} the coefficient function in the transformed equation (4) for 2 < α < 4, q(x) ≤ λx^{−α/2}, so by Sturm Comparison Theorem between two consecutive zeros a_{k_{0}+1} and a_{k_{0}} of y(x), ({a_{k}} denotes the decreasing sequence of consecutive zeros of y(x)) there exists at least one zero x_{n_{0}} such that a_{k_{0}+1} < x_{n_{0}} < a_{k_{0}}. Repeating this procedure to all pairs of consecutive zeros a_{a_{0}+j} and a_{k_{0}+j−1}, we obtain another zero x_{n_{0}+n_{j}−1}, n_{j} ≥ j such that a_{k_{0}+j} < x_{n_{0}+n_{j}−1} < a_{k_{0}+j−1}.

We note that for the segment of the solution curve Γ_{k} = {(x, y(x)) : a_{k+1} ≤ x ≤ a_{k}} its arc length L(Γ_{k}) satisfies

(13)  2 |y (s_{k})| ≤ L (Γ_{k}) ≤ 2 |y (s_{k})| + (a_{k} − a_{k+1}) ,

where y(x) attains its extrema s_{k} between a_{k+1} and a_{k}, i.e. y′(s_{k}) = 0. Piecing together the segments Γ_{k}, we note that the arc length of the Graph of solution curve satisfies

(14)  L_{G}(y) = ∑_{k=0}^{∞} L(Γ_{k}) + arc length {(x, y(x) : a_{0} ≤ x ≤ 1} .

the last term in (14) is a positive constant depending on the solution y(x) which we denote by M_{0}. From (13) and (14), we obtain

(15)  2 ∑_{k=0}^{∞} |y(s_{k})| ≤ L_{G}(y) ≤ 2 ∑_{k=0}^{∞} |y(s_{k})| + M_{0} + a_{0},

since ∑_{k=0}^{∞} (a_{k} − a_{k+1}) = a_{0}. Estimate (15) allows us to conclude that the solution y(x) is rectifiable or unrectifiable depending on whether the series ∑_{k=0}^{∞} |y(s_{k})| is convergent or divergent.

Writing k = k_{0} + j, we obtain the estimate

(16)  a_{k} = a_{k_{0}+j} < x_{n_{0}+n_{j}−1} ≤ x_{n_{0}+j−1} = k_{α} (\frac{1}{n_{0} + k − k_{0} − 1})

where k_{α} = (\frac{\sqrt{λ}}{σπ})^{1/σ} , σ = (\frac{α}{2} − 1). Since |y(x)| ≤ cx^{α/4} for x ∈ (0, b] and a_{k+1} < s_{k} < a_{k},

we find

(17)  ∑_{k=0}^{∞} |y(s_{k})| ≤ ∑_{k=0}^{∞} s_{k}^{α/4} ≤ c ∑_{k=0}^{∞} a_{k}^{α/4}.

Now consider α/4σ as a function of α which for α > 2 is decreasing in α and equals to 1 at α = 4. Combining (16)and (17), we obtain for k_{1} ≥ n_{0} − k_{0} − 1

∑_{k=k_{1}}^{∞} |y(s_{k})| ≤ ck_{α}^{α/4} ∑_{k=1}^{∞} (\frac{1}{k} )^{α/4σ} < ∞,

since α/4σ > 1 for 2 < α < 4. This proves (a).

Returning to the case when α ≥ 4, we have q(x) ≥ λx^{−α/2} so we can apply Sturm’s Comparison theorem to equations (4) and (11) and conclude that between two zeros (12) of solutions of (11), x_{k_{0}+1} < x_{k_{0}}, there exists at least one zero a_{i_{0}}, i.e. x_{k_{0}+1} < a_{i_{0}} < x_{k_{0}}. Repeating this process to all pairs of consecutive zeros, x_{k_{0}+k} < x_{k_{0}+k−1}, we obtain a_{i_{0}+i_{k}} satisfying x_{k_{0}+k} < a_{i_{0}+i_{k}} < x_{k_{0}+k−1} where i_{k} ≥ k. Hence by (12)

(18)  K_{α} (\frac{1}{k_{0} + k} )^{1/σ} = x_{k_{0}+k} < a_{i_{0}+i_{k}} < a_{i_{0}+k}, k_{α} = (\frac{\sqrt{λ}}{σπ})^{1/σ} .

Let w(x) be a linearly independent solution of y(x) chosen such that the Wronskian W (y, w)(x) = y(x)w′(x) − w(x)y′(x) is a constant, say W (y, w) ≡ 1. Evaluating W (y, w)(x) at x = s_{k} where y′(s_{k}) = 0, we obtain by boundary layer condition (II) when applied to w(x) the following estimate :

(19)  |y (s_{k})| ≥ \frac{1}{|w′(s_{k})|} ≥ \frac{1}{c}s_{k}^{α/4} ≥ \frac{1}{c}a_{k+1}^{α/4}

EJQTDE, 2007 No. 20, p. 6 Combining (18) and (19), we have

(20)  ∑_{k=i_{0}+1}^{∞} |y (s_{k})| ≥ K_{α}^{α/4}c^{−1} ∑_{k=i_{0}+1}^{∞} (\frac{1}{k_{0} + k} )^{α/4σ} = ∞,

since α/4σ ≤ 1 for α ≥ 4. The divergence of the series in (20) shows that solution g(x) is unrectifiable. This completes the proof of (b).

§3. In this section, we show that our Theorem can be further extended to give a somewhat more general result for the harmonic oscillator equation:

(21)  y′′ + f(x)y = 0,  x ∈ (0, 1]

where f(x) > 0 and f(x) ∼ λx^{−α}, λ > 0, α > 2, as x → 0, i.e. lim_{x→∞} x^{α}f(x) = λ. If f(x) ∈ C^{2}(0, 1] and satisfies

(22)  lim_{ε→0} ∫_{ε}^{1} f^{− \frac{1}{4}} (x) |(f^{− \frac{1}{4}} (x))′′| dx < ∞,

then conclusion of the Theorem remains valid.

Given the asymptotic behavior f(x) ∼ λx^{−α}, λ > 0, α > 2 and condition (22), we can employ the WKB type asymptotic integration formula for solutions of (21) due to Wintner [9] see Coppel [4; p.122], which states that lim_{ε→0} ∫_{ε}^{1} \sqrt{f(x)}dx = ∞ plus condition (22) imply that all solutions of (21) satisfy the asymptotic formula:

(23)  y(x) = f^{− \frac{1}{4}} (x) \{c_{1} sin ∫_{x}^{1} \sqrt{f(ξ)}dξ + c_{2} cos ∫_{x}^{1} \sqrt{f(ξ)}dξ + o(1)\}

and

(24)  y′(x) = f^{\frac{1}{4}} (x) \{c_{1} sin ∫_{x}^{1} \sqrt{f(ξ)}dξ + c_{2} cos ∫_{x}^{1} \sqrt{f(ξ)}dξ + o(1)\} ,

which together imply y(x) = 0 (f^{− \frac{1}{4}} (x)) and y′(x) = 0 (f^{\frac{1}{4}} (x)) as x → 0. The asymptotic behaviour f(x) ∼ λx^{−α} shows that for 0 < ε < λ, there exists δ > 0 and 0 < b < 1 such that

(25)  0 < λ − ε < f(x)x^{α} < λ + ε for all x ∈ (0, b].

EJQTDE, 2007 No. 20, p. 7 Since α > 2, (25) yields lim_{ε→0} ∫_{ε}^{1} \sqrt{f(x)}dx = ∞ so that subject to (22) formula (23) and (24) are applicable. We note that zeros a_{k} of y(x) occur at:

∫_{a_{k}}^{1} \sqrt{f(ξ)}dξ = kπ

which by (25) gives the estimate

(26)  \sqrt{λ − ε} ∫_{a_{k}}^{1} \frac{dξ}{ξ^{α/2+1=σ}} ≤ kπ or \frac{\sqrt{λ − ε}}{σ} \{a_{k}^{σ} − 1\} ≤ kπ

where σ = \frac{α}{2} − 1 > 0. Denote λ_{1}^{−1} = \sqrt{λ − ε}/σ. It follows from (26) that

(27)  a_{k} ≥ (\frac{1}{λ_{1}kπ + 1})^{\frac{1}{σ}} ; k = 1, 2, · · · · · ·

Let w(x) be another solution of (21) which is linearly independent from y(x) and can be chosen so that the Wronskian y(x)w′(x)−y′(x)w(x) ≡ 1. As before, denote {s_{k}} be sequence of consecutive extrema of y(x), i.e. y′(s_{k}) = 0, a_{k+1} < s_{k} < a_{k}. Since w′(x) also satisfies (24), so by (25)

(28)  |y(s_{k})| ≥ \frac{1}{|w′(s_{k})|} ≥ \frac{1}{f^{1/4}(s_{k})} > \frac{s_{k}^{α/4}}{(λ + ε)^{1/4}} = λ_{2}s_{k}^{α/4},

where λ_{2} = (λ + ε)^{−1/4}. Since s_{k} > a_{k+1}, we obtain form (27) and (28)

(29)  ∑_{k=1}^{∞} |y(s_{k})| ≥ λ_{2} ∑_{k=1}^{∞} a_{k+1}^{α/4} ≥ λ_{2} ∑_{k=1}^{∞} (\frac{1}{λ_{1}kπ + 1})^{α/4σ} .

Again we note that α/4σ = α/(2α − 4) is decreasing in α and equals to 1 when α = 4. So if α ≥ 4 we have α/4σ ≤ 1. Thus the infinite series appeared as the last term in (29) diverges, so does ∑_{k=1}^{∞} |y(s_{k})|. Returning to the estimate (15) which related the arc length of the solution curve y(x) to ∑_{k=1}^{∞} |y(s_{k})|, the divergence of ∑_{k=1}^{∞} |y| proves that y(x) has infinite arc length, i.e. y(x) is unrectifiable.

The case 2 < λ < 4 is similar. We leave the details to the interested reader and therefore conclude that solution curves of (21) have finite arclength. Hence we have proved Theorem 2. Suppose that f(x) ∈ C^{2}(0, 1] and f(x) > 0. If f(x) ∼ λx^{−α}, λ > 0, α > 2 as x → 0 and f(x) satisfies (22), thus

(a) for α < 4, all solution of (21) are rectifiable oscillatory; and

(b) for α ≥ 4, all solutions of (21) are unrectifiable oscillatory.

§4. In this last section, we give another example of unrectifiable oscillation and several remarks concerning results discussed in this paper. Example. Consider a special case of equation (21)

(30)  y′′ + x^{−4} exp(2/x)y = 0,

where the coefficient f(x) is highly singular at x = 0, but is not of Euler type, i.e. f(x) is not asymptotic to a negative power of x. Furthermore, f(x) = x^{−4} exp (2/x) satisfies

(i) ∫_{ε}^{1} \sqrt{f(x)}dx = exp(1/ε) − e → ∞ as ε → 0

(ii) lim_{ε→0} ∫_{ε}^{1} f^{− \frac{1}{4}} (x) |(f^{− \frac{1}{4}} (x))′′| dx = lim_{ε→0} ∫_{ε}^{1} \frac{1}{4x^{3}} exp (−\frac{1}{x}) dx < ∞,

so condition (22) of Wintner’s asymptotic formula is satisfied. Hence all solutions of (30) satisfy the boundary layer conditions (I) and (II) introduced in Pasic [8].

Once again let w(x) be the solution of (30) linearly independent of y(x) such that the Wronkian W (y, w)(x) = (yw′ − wy′)(x) ≡ 1. Denote {x_{n}} the sequence of consecutive zeros of y(x), x_{n} → 0 as n → ∞ and {s_{n}} the corresponding consecutive extrema {s_{n}}, i.e. y′(s_{n}) = 0, such that x_{n+1} < s_{n} < x_{n} (the uniqueness of s_{n} between the two zeros x_{n+1} and x_{n} is guaranteed by the concavity of y(x), i.e. y′′(x) ≤ 0).

Using the asymptotic formula (23), (24) of any solution y(x) and w(x) of (30), we can determine its extrema s_{n} by the formula

(31)  e^{\frac{1}{sn}} − e = (n + \frac{1}{2}) π or s_{n} = [log (e + (n + \frac{1}{2}) π)]^{−1}

We can now estimate |y(s_{n})| by using (31) to obtain

(32) \begin{aligned}|y(s_{n})| = \frac{1}{|w′(s_{n})|} & ≥ s_{n} exp (−\frac{1}{2s_{n}} ) = [log (e + n + \frac{1}{2}) π]^{−1} [e + (n + \frac{1}{2}) π]^{− \frac{1}{2}} \\ & ≥ [e + (n + \frac{1}{2}) π]^{−1} ≥ [(n + 2)π]^{−1} . \end{aligned}

since log x ≤ \sqrt{x} for large value of x and e + \frac{1}{2}π ≤ 2π. Summing up the terms |y(s_{n})| in (32), we find

∑_{n=1}^{∞} |y(s_{n})| ≥ \frac{1}{π} ∑_{n=3}^{∞} \frac{1}{n} = ∞,

which proves that y(x) is unrectifiable oscillatory. Remark 1. Condition (22) required for Wintner’s asymptotic formula has another form given by Atkinson [3];

(30)  lim_{ε→0} ∫_{ε}^{1} f^{− \frac{3}{2}} (x)|f′′(x)|dx < ∞

which may be easier to apply. (See Coppel [4; p.122]). Suppose that f(x) ∼ x^{−α}, α > 2, as x → 0 and f(x) is sufficiently smooth such that its second derivative satisfies f′′(x) ∼ x^{−α−2}. Then it is easy to see that

lim_{ε→0} ∫_{ε}^{1} f^{−3/2}|f′′|x = lim_{ε→0} (\frac{1 − ε^{α/2−1}}{\frac{α}{2} − 1} ) = \frac{2}{α − 2} < ∞

which is of course finite because α > 2. Remark 2. The proof can be modified if we assume instead of f(x) ∼ λx^{−α}, λ > 0, α > 2, as x → 0 the inequality.

m_{1}x^{−α} ≤ f(x) ≤ m_{2}x^{−α}, x ∈ (0, b]

where m_{1}, m_{2}, and b are positive constants and 0 < b < 1. Remark 3. It may be of interest to digress to a discussion on the oscillation criteria of equations (21) over the finite interval (0, 1]. If we are concerned with the semi-infinite interval [1, ∞), then the standard Fite-Wintner oscillation criterion states that ∫_{1}^{∞} x^{μ}a(x)dx = ∞ for any μ < 1 implies oscillation of (21). However the condition that ∫_{0}^{1} \sqrt{a(x)}dx = ∞ imposed in Theorem 2 does not always ensure oscillation since the Euler equation y′′ + \frac{1}{4}x^{−2}y = 0 has y(x) = \sqrt{x} as a nonoscillatory solution. The corresponding Fite-Wintner oscillation criterion for the finite interval is ∫_{0}^{1} x^{2−μ}a(x)dx = ∞ for any μ < 1, which is satisfied for a(x) = λx^{−α}, λ > 0, α > 2.

EJQTDE, 2007 No. 20, p. 10 Remark 4. Oscillation of the harmonic oscillator (21) was traditionally discussed for the semi-infinite interval where the independent variable x relates to time and the dependent variable y relates to the travelling waves. However for studies in some areas such as nuclear physics, the independent variable x relates to radial distance from the centre of the nuclei and the dependent variable y relates to nuclear charge of the atom(s). How does one interpret the physical meaning of unrectifiable oscillation in physical problems is certainly a most interesting question. Remark 5. The characteristic exhibited by solution y(x) in (3) is interesting because it arises from a simple linear differential equations whilst its solution curve over a finite interval has infinite arclength. Such curves are also known as fractals which normally associated with chaos in nonlinear dynamical systems. See e.g., Le Mehaute [7]. Addison [2] for further discussions on fractal geometry and chaos.

References

[1] T.A. Apostol, \underline{Mathematical Analysis}, Addison-Wesley, Reading, Massacdusetts, 1957.

[2] P.S. Addison, \underline{Fractals and Chaos - An illustrated Course}, Institute of Physics Publishing, Bristol, 1997.

[3] F.V. Atkinson, “Asymptotic formulae for linear oscillations”, Proc. Glasgow Math. Assoc.,3 (1956-8), 105-111.

[4] W.A. Coppel, \underline{Stability and Asymptotic Behavior of Differential Equations}, D.C. Heath, Boston, 1965.

[5] R.L. Devancy, \underline{A first Course in Chaotic Dynamical Systems}, Addison-Wesley, Reading, Massacdusetts, 1992.

[6] P. Hartman, \underline{Ordinary Differential Equations}, John Wiley and Sons, New York, 1974.

[7] A.Le. Mehaute, \underline{Fractal Geometries - Theory and Applications},

CRC Press Inc., Boca Raton, 1991.

EJQTDE, 2007 No. 20, p. 11 [8] M. Pasic, “Rectifiable and unrectifiable oscillations for a class of second order linear

differential equations of Euler type”, J. Math. Anal., & Appl. (to appear),

[9] A. Wintner, “Asymptotic integration of the adiabatic oscillator”, Amer. J. Math., 69 (1947), 251-272.

(Received April 16, 2007)