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Succeeding in advanced mathematics examinations requires more than just memorizing formulas; it demands a deep, intuitive understanding of how differential equations and variational problems behave. This text-based course is designed to bridge the gap between basic calculus and the rigorous problem-solving strategies required for competitive exams like the CSIR NET. You will transition from a foundational understanding of mathematical definitions to confidently solving complex boundary value problems and extremizing functionals. Through clear written explanations, step-by-step derivations, and structured practice problems, you will build the analytical skills necessary to tackle both theoretical and applied questions with precision. What you'll learn: Learn the fundamental classification, existence, and uniqueness theorems of ordinary differential equations; Solve first-order and higher-order linear differential equations using systematic analytical methods; Apply the Euler-Lagrange equation to find extremals of functionals in the Calculus of Variations; Master variational problems with constraints using the method of Lagrange multipliers; Classify and solve second-order partial differential equations, including wave, heat, and Laplace equations; Analyze qualitative behaviors of systems, including phase portraits and stability analysis. This course begins with essential terminology, basic definitions, and first-order equations before advancing to higher-order systems, partial differential equations, and variational calculus. Each module focuses on core theory followed by rigorous, step-by-step written examples modeled after exam-style questions. This course is designed for mathematics students, graduates, and aspirants preparing for competitive examinations who want a structured, text-only guide to mastering differential equations and the calculus of variations from the ground up. No advanced prerequisites are required, though a basic familiarity with single-variable calculus is helpful. Start building your mathematical problem-solving toolkit today.