A STUDY OF LAPLACE TRANSFORMS IN ORDINARY DIFFERENTIAL EQUATIONS

International Journal of Computer Science (IJCS) Published by SK Research Group of Companies (SKRGC)

Abstract

The Laplace transform is one of the most powerful analytical tools for solving linear ordinary differential equations (ODEs) with constant coefficients, particularly initial value problems arising in engineering and applied science. This paper presents a systematic study of the Laplace transform method, beginning with its formal definition, existence conditions, and fundamental operational properties, followed by a compiled table of standard transform pairs. A step-by-step methodology for converting a linear ODE into an algebraic equation in the complex frequency domain, solving for the transformed variable, and recovering the time-domain solution through inverse transformation and partial-fraction expansion is developed and illustrated with worked examples drawn from mechanical vibration and electrical circuit problems. Numerical simulations are used to visualize the underdamped, critically damped, and overdamped responses of a second-order system, the unit-step response of a forced system, and the transient current in a series RL circuit. A comparative analysis contrasts the Laplace transform approach with classical methods such as undetermined coefficients and variation of parameters, highlighting its advantages in handling discontinuous and impulsive forcing functions and in directly incorporating initial conditions. The results confirm that the Laplace transform offers a computationally efficient and conceptually unifying framework for analyzing linear time-invariant systems, with direct relevance to control theory, circuit analysis, and structural dynamics.

References

1. E. Kreyszig, Advanced Engineering Mathematics, 10th ed. Hoboken, NJ, USA: Wiley, 2011.
2. W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 11th ed. Hoboken, NJ, USA: Wiley, 2017.
3. M. R. Spiegel, Laplace Transforms, Schaum's Outline Series. New York, NY, USA: McGraw-Hill, 1965.
4. K. Ogata, Modern Control Engineering, 5th ed. Upper Saddle River, NJ, USA: Prentice Hall, 2010.
5. C. K. Alexander and M. N. O. Sadiku, Fundamentals of Electric Circuits, 6th ed. New York, NY, USA: McGraw-Hill, 2016.
6. D. G. Zill, A First Course in Differential Equations with Modeling Applications, 11th ed. Boston, MA, USA: Cengage Learning, 2017.
7. R. N. Bracewell, The Fourier Transform and Its Applications, 3rd ed. New York, NY, USA: McGraw-Hill, 2000.
8. N. S. Nise, Control Systems Engineering, 7th ed. Hoboken, NJ, USA: Wiley, 2015.

           

Keywords

Laplace transform, ordinary differential equations, initial value problems, inverse Laplace transform, s-domain, transfer function, partial fraction expansion.