Laplace Transforms : Definition of Laplace transform, linearity property–Piecewise continuous function. Existence of Laplace transform, Functions of exponential order and of class A. First and second shifting theorems of Laplace transform, Change of scale property- Laplace transform of derivatives, Initial value problems, Laplace transform of integrals, Multiplication by t, Division by t, Laplace transform of periodic functions and error function, Beta function and Gamma functions. Definition of Inverse Laplace transform, Linearity property, First and second shifting theorems of inverse Laplace transform, Change of scale property, Division by p, Convolution theorem, Heaviside’s expansion formula (with proofs and applications).
Applications of Laplace Transforms : Applications of Laplace Transforms to the solution of ordinary differential equations with constant coefficients and variable coefficients, Simultaneous ordinary differential equations, Second order partial differential equations (Heat equation, Wave equation and Laplace equation)