Theory of Third-Order Differential Equations
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This book discusses the theory of third-order differential equations. Most of the results are derived from the results obtained for third-order linear homogeneous differential equations with constant coefficients. M. Gregus, in his book written in 1987, only deals with third-order linear differential equations. These findings are old, and new techniques have since been developed and new results obtained.
Chapter 1 introduces the results for oscillation and non-oscillation of solutions of third-order linear differential equations with constant coefficients, and a brief introduction to delay differential equations is given. The oscillation and asymptotic behavior of non-oscillatory solutions of homogeneous third-order linear differential equations with variable coefficients are discussed in Ch. 2. The results are extended to third-order linear non-homogeneous equations in Ch. 3, while Ch. 4 explains the oscillation and non-oscillation results for homogeneous third-order nonlinear differential equations. Chapter 5 deals with the z-type oscillation and non-oscillation of third-order nonlinear and non-homogeneous differential equations. Chapter 6 is devoted to the study of third-order delay differential equations. Chapter 7 explains the stability of solutions of third-order equations. Some knowledge of differential equations, analysis and algebra is desirable, but not essential, in order to study the topic.
have G2 + 4H 3 > 0. Thus 2c ≤ ab implies that (1.5) is oscillatory. (v) If (1.5) admits an oscillatory solution and G > 0, then (1.7) admits two imaginary roots α + iβ and α − iβ and a real negative root γ . As α + iβ + α − iβ + γ = 0, then α > 0. Clearly a a a e(α− 3 )t cos βt, e(α− 3 )t sin βt, e(γ − 3 )t is a basis of the solution space of (1.5). We show that nonoscillatory solutions of (1.5), along with the trivial solution, form a one-dimensional subspace of the solution space of (1.5).
− N2 ). r(θ ) Hence (2.58) yields t z(t) > −αMr(t) β 1 r(s) β s αM dθ ds = − r(t) r(θ ) 2 t β ds r(s) 2 . The proof is complete. Theorem 2.2.4 If a(t) ≤ α < 0 for t ≥ σ and (2.1) is nonoscillatory, then (2.1) admits a nonoscillatory solution which tends to zero as t → ∞. Proof Clearly (2.1) is oscillatory, if and only if (2.2) is oscillatory. From Lemma 1.5.11, it follows that (2.49) is oscillatory. Suppose that z1 (t) is an oscillatory solution of (2.49). By Lemma 2.2.2, there exists
(2.64) for every n and tn lim n→∞ For M = − 16u such that tn β β 2 (β)W (β)z′′ (β) αλ3 u(s) ds r(s) 2 2 u(s) ds r(s) r(tn ) ru′′ + qu (tn ) = µ. u(tn ) u2 > 0, there exists a real number N > β (by Lemma 2.2.4) ≤ u2 (β) tn β ds r(s) 2 ≤− 2u2 (β)z(tn ) , αMr(tn ) tn ≥ N. This, in turn, because of (2.64) and (2.62), implies that tn β u(s) ds r(s) 2 r(tn ) ru′′ + qu 2u2 (β)W (β)z′′ (β) 1 (t ) ≤ − = . n u(tn ) 8 u2 αMλ3 72 2 Behaviour of Solutions of Linear Homogeneous
satisfies limt→∞ x(t) = 0. Theorems 2.7.3–2.7.5 yield the following corollary: Corollary 2.7.9 If (2.127) holds, then (2.120) is nonoscillatory. Corollary 2.7.10 If (2.131) holds, then (2.123) is nonoscillatory and Eq. (2.124) is nonoscillatory. Now, we consider the second-order linear homogeneous equation (2.94), where b is a continuous function for t ≥ σ with either b(t) > 0 or b(t) < 0 for t ≥ σ . Let h(t) be a positive solution of (2.94) on [t0 , ∞), t0 ≥ σ . Then the third-order differential
it is oscillatory in the sense of Definition 1.1.2. These two definitions are equivalent for solutions of linear homogeneous secondorder differential equations of the form ′ r(t)x ′ + p(t)x = 0, but not equivalent for a function x ∈ C([σ, ∞), R). We may note that the equation x ′′ = t 2 30t 2 − 1 sin 1 1 − 10t 3 cos + 21t 5 , t t t >0 admits a solution x(t) = t 6 (sin 1t + 2t ), which has an infinite number of zeros in (0, 1], a finite number of zeros in [1, 2] and it is positive on (2, ∞).