Linear differential equations with variable coefficients arise naturally in various scientific and engineering contexts, including fluid dynamics, quantum mechanics, population models, and control systems. Unlike constant coefficient equations, these equations often resist closed-form solutions, making qualitative analysis essential. This paper presents a comprehensive study of the qualitative behaviour of linear differential equations with variable coefficients. It explores existence and uniqueness theorems, stability analysis, oscillatory behaviour, asymptotic properties, and transformation techniques. The paper emphasizes understanding solution behaviour without explicit solutions, using analytical tools such as comparison theorems, Lyapunov methods, and phase analysis. Applications and illustrative examples are also discussed to highlight practical relevance.

Keywords: Stability Analysis, Oscillatory Behaviour, Asymptotic Behaviour, Lyapunov Stability, Sturm Comparison Theorem