Engineering Mathematics-II

Unit I

Ordinary Differential Equations of Higher Order: 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, Applications in solving engineering problems.

Unit II

Laplace Transform: Laplace transform, Existence theorem, Properties, Laplace transform of derivatives and integrals, Unit step function, Laplace transform of periodic functions, Inverse Laplace transform, Convolution theorem, Application to solve ordinary and simultaneous differential equations.

Unit III

Sequence and Series: Definition of sequence and series with examples, Convergence of series, Tests for convergence — Ratio test, D'Alembert's test, Raabe's test, Comparison test. Fourier series, Half range Fourier sine and cosine series.

Unit IV

Complex Variable — Differentiation: Functions of complex variable, Limit, continuity and differentiability, Analytic functions, Cauchy-Riemann equations (Cartesian and Polar form), Harmonic functions, Methods to find analytic functions, Milne-Thompson method, Conformal mapping, Mobius transformation and their properties.

Unit V

Complex Variable — Integration: Complex integration, Cauchy integral theorem, Cauchy integral formula, Taylor's and Laurent's series, Singularities and their classification, Zeros of analytic functions, Residues, Cauchy's Residue theorem and its applications.