Unit -1: Ordinary Differential Equation of Higher Order 8
Linear differential equation of nth order with constant coefficients, Simultaneous linear differential equations, Second order linear differential equations with variable coefficients, Solution by changing independent variable, Method of variation of parameters, Cauchy-Euler equation, Application of differential equations in solving engineering problems.
Unit-2: Laplace Transform 10
Laplace transform, Existence theorem, Properties of Laplace Transform, Laplace transform of derivates and integrals, Unit step function, Laplace transform of periodic function, Inverse Laplace transform, Convolution theorem. Application of Laplace Transform to solve ordinary differential equations and simultaneous differential equations.
Unit-3: Sequence and Series 8
Definition of Sequence and series with examples, Convergence of series, Tests for convergence of series, Ratio test, D’ Alembert’s test, Raabe’s test, Comparison test. Fourier series, Half range Fourier sine and cosine series.
Unit-4: Complex Variable–Differentiation 8
Functions of complex variable, Limit, Continuity and differentiability, Analytic functions, Cauchy- Riemann equations (Cartesian and Polar form), Harmonic function, Method to find Analytic functions, Milne’s Thompson Method, Conformal mapping, Mobius transformation and their properties.
Unit-5: Complex Variable –Integration 8
Complex integration, Cauchy- Integral theorem, Cauchy integral formula, Taylor’s and Laurent’s series, singularities and its classification, zeros of analytic functions, Residues, Cauchy’s Residue theorem and its application.