Solutions Of Second-order Linear Equations Research Articles

Asymptotic equivalence of linear stochastic Itô systems and oscillation of solutions of linear second-order equations

We obtain conditions for the asymptotic equivalence of linear stochastic and deterministic systems and analyze the oscillation of solutions of the Ito stochastic equation of the second order of the form \(\ddot x + (p(t) + q(t)\dot W(t))x = 0\) on the half-line.

Ordinary Differential Equations with Modern Applications.

Elementary methods: first-order differential equations. Linear differential equations. Linear systems. The Laplace transform. Series solutions of second-order linear equations. Boundary value problems. Numerical solutions of differential equations. Nonlinear differential equations and systems.

Asymptotic solutions of second-order linear equations with three transition points

A uniformly valid asymptotic expansion is obtained for the regular solution of a class of second-order linear differential equations with three transition points-a turning point and two regular singular points. The solution is found by matching three different solutions obtained using the Langer Transformation. The matching yields the eigenvalues and the eigenfunctions.

Increasing solutions of linear second-order equations with nonnegative characteristic form

In a layer H{0< t ≤ T, x e Rn we consider a linear second-order parabolic equation that degenerates on an arbitrary subset ¯H. It is assumed that the coefficient of the time derivative has a zero of sufficiently high order on the hyperplane t=0; as a consequence, the Cauchy problem will be unsolvable. The exact bounds are obtained of the permissible growth of the sought-for function when ¦x¦→ ∞, ensuring a single-valued solution of the problem without initial data.

Approximate oscillatory solution of second-order linear equations, with application to a Bessel equation

Approximate oscillatory solution of second-order linear equations, with application to a Bessel equation