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In this paper, we are connected with oscillation of third-order linear neutral delay difference equation of the form Δ ( c 2 ( σ ) Δ ( c 1 ( σ ) Δ h ( σ ) ) ) + p ( σ ) x ( σ − ς ) = 0 , σ ≥ σ 0 > 0 , where h ( σ ) = x ( σ ) + q ( σ ) x ( σ − η ) . Here c 1(σ) and c 2(σ) are the sequence of positive integers, p(σ) and q(σ) are positive and real sequences such that q(σ) ≥ q 0 > 1 and p(σ) ≠ 0, ς, η are positive integers, such that ς > η.

In this manuscript, we discuss the Tarig transform for homogeneous and non-homogeneous linear differential equations. Using this Tarig integral transform, we resolve higher-order linear differential equations, and we produce the conditions required for Hyers–Ulam stability. This is the first attempt to use the Tarig transform to show that linear and nonlinear differential equations are stable. This study also demonstrates that the Tarig transform method is more effective for analyzing the stability issue for differential equations with constant coefficients. A discussion of applications follows, to illustrate our approach. This research also presents a novel approach to studying the stability of differential equations. Furthermore, this study demonstrates that Tarig transform analysis is more practical for examining stability issues in linear differential equations with constant coefficients. In addition, we examine some applications of linear, nonlinear, and fractional differential equations, by using the Tarig integral transform.