Solutions of disconjugate third-order differential equations

Liouvillian solutions of third order differential equations

The Kovacic algorithm and its improvements give explicit formulae for the Liouvillian solutions of second order linear differential equations. Algorithms for third order differential equations also exist, but the tools they use are more sophisticated and the computations more involved. In this paper we refine parts of the algorithm to find Liouvillian solutions of third order equations. We show...

Oscillation of Third-order Functional Differential Equations

The aim of this paper is to study oscillatory and asymptotic properties of the third-order nonlinear delay differential equation (E) ˆ a(t) ˆ x ′′(t) ̃ γ ̃ ′ + q(t)f(x [τ (t)]) = 0. Applying suitable comparison theorems we present new criteria for oscillation or certain asymptotic behavior of nonoscillatory solutions of (E). Obtained results essentially improve and complement earlier ones. Variou...

ON TE EXISTENCE OF PERIODIC SOLUTION FOR CERTAIN NONLINEAR THIRD ORDER DIFFERENTIAL EQUATIONS

Rational solutions of first-order differential equations∗

We prove that degrees of rational solutions of an algebraic differential equation F (dw/dz,w, z) = 0 are bounded. For given F an upper bound for degrees can be determined explicitly. This implies that one can find all rational solutions by solving algebraic equations. Consider the differential equation F (w′, w, z) = 0, (w′ = dw/dz) (1) where F is a polynomial in three variables. Theorem 1 For ...

Stability and Boundedness of Solutions of Nonlinear Vector Differential Equations of Third Order

The paper studies the equation ... X + Ψ(Ẋ)Ẍ + Φ(X)Ẋ + cX = P (t) in two cases: (i) P (t) ≡ 0, (ii) P (t) 6= 0. In case (i), the global asymptotic stability of the solution X = 0 is studied; in case (ii), the boundedness of all solutions is proved.