Applications of Laplace Transform in Engineering and Applied Mathematics

The Laplace transform is one of the most powerful integral transforms in applied mathematics, converting linear ordinary differential equations with constant coefficients into algebraic equations that are far easier to manipulate. This paper presents a comprehensive study of the Laplace transform, its fundamental properties, and its applications across electrical engineering, mechanical vibration analysis, control systems, and signal processing. The definition of the transform, together with linearity, shifting, differentiation, integration, and convolution properties, is developed and organised into reference tables. These tools are then applied to solve initial value problems for spring-mass-damper systems and series RLC circuits without first finding the general homogeneous solution, to analyse system stability through pole locations in the complex s-plane, and to characterise linear time-invariant systems using transfer functions and impulse response. Worked numerical examples, comparative tables contrasting the Laplace approach with classical methods, and simulation graphs of transient and steady-state responses are provided throughout. The paper concludes that the Laplace transform, by converting differentiation into multiplication and convolution into multiplication in the transform domain, remains an indispensable analytical technique for engineers and applied mathematicians dealing with linear systems subject to arbitrary, including discontinuous and impulsive, forcing functions