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In this paper, we establish the Shehu transform expression for homogeneous and non-homogeneous linear differential equations. With the help of this new integral transform, we solve higher order linear differential equations in the Shehu sense. Moreover, this paper introduced a new concept to find the Hyers-Ulam stability of the differential equation. This is the first attempt to use the Shehu transform to prove the Hyers-Ulam stability of the differential equation. Finally, the applications and remarks are discussed to demonstrate our strategy. Applications of the Shehu transform to fractional differential equations, Newton’s law of cooling, and free undamped motion are also discussed.