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A detailed mathematical analysis of differential equations and their applications. It covers various concepts such as first-order linear differential equations, second-order linear differential equations, and their solutions. The document delves into the properties of these equations, including their general solutions, particular solutions, and the behavior of the solutions. It also explores the applications of differential equations in various fields, such as physics, engineering, and biology. The mathematical expressions and notations used suggest that this document is likely intended for an audience with a strong background in advanced mathematics, such as university-level students or researchers in the field of applied mathematics or mathematical physics.
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