A book that goes very much in depth, explaining and reasoning out every topic is the book that I prefer. The topics I require are: Unit-I: First order Differential Equations (10) Formation – Variables seperable – Homogeneous, non Homogeneous, Linear and Bernoulli equations. Exact equations - Applications of first order differential equations – Orthogonal Trajectories, Newton’s law of cooling, law of natural growth and decay. Unit-II: Higher order Differential Equations (12) Complete solutions - Rules for finding complementary function - Inverse operator - Rules for finding particular integral - Method of variation of parameters - Cauchy’s and Legendre’s linear equations - Simultaneous linear equations with constant coefficients - Applications of linear differential equations to Oscillatory Electrical circuits L-C, LCR – Circuits - Electromechanical Analogy. Unit-III: Mean Value Theorems (08) Rolle’s, Lagrange’s and Cauchy’s mean value theorems. Taylor’s and Maclaurin’s theorems and applications (without proofs). Unit-IV: Infinite Series (12) Definitions of convergence, divergence and oscillation of a series - General properties of series - Series of positive terms - Comparison tests - Integral test - D’ Alembert’s Ratio test - Raabe’s test - Cauchy’s root test - Alternating series - Leibnitz’s rule - Power series - Convergence of exponential, Logarithmic and binomial series (without proofs). Unit-V: Linear Algebra (12) Rank of a Matrix – Elementary Transformations – Echelon form - Normal form (self study). Consistency of Linear system of equations A X = B and A X = 0. Eigen Values and Eigen Vectors – Properties of eigen values(without proofs) – Cayley – Hamilton theorem (Statement only without proof) – Finding inverse and powers of a square matrix using Cayley– Hamiton theorem – Reduction to diagonal form – Quadratic form - Reduction of Quadratic form into canonical form – Nature of quadratic forms. Could be 1 book or more. But like I said I prefer books that go very much in detail like Calculus by G.B Thomas. What a genius of a book!