Abstract
It is shown that the differential equation
u
(n) = p(t)u,
where n ≥ 2 and p : [a, b] → ℝ is a summable function, is not conjugate in the segment [a, b], if for some l ∈ {1, . . . , n – 1}, α ∈]a, b[ and β ∈]α, b[ the inequalities
hold.
The purpose of this paper is to investigate the use of exponential Chebyshev (EC) collocation method for solving systems of high-order linear ordinary differential equations with variable coefficients with new scheme, using the EC collocation method in unbounded domains. The EC functions approach deals directly with infinite boundaries without singularities. The method transforms the system of differential equations and the given conditions to block matrix equations with unknown EC coefficients. By means of the obtained matrix equations, a new system of equations which corresponds to the system of linear algebraic equations is gained. Numerical examples are given to illustrative the validity and applicability of the method.
AbstractAn efficient condition is established ensuring that on any interval of length ω,
any nontrivial solution of the equation ${u^{\prime\prime}=p(t)u}$ has at most one zero.
Based on this result, the unique solvability of a periodic boundary value problem is studied.