TABLE OF CONTENT TITLE PAGE i DECLARATION i CERTIFICATION ii DEDICATION iii ACKNOWLEDGEMENT iv TABLE OF CONTENTS vi ABSTRACT viii 1.0 INTRODUCTION 1 1.1 Background Information . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Definition of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Statement of The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Aim and Objective of The Study . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Significance of The Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.6 Scope of Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.0 LITERATURE REVIEW 7 2.1 Application of Laplace Transform to Differential Equations . . . . . . . . . . . . 12 3.0 RESEARCH METHODOLOGY 15 3.1 Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Properties of Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Laplace Transform of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.4 LAPLACE TRANSFORM OF CONVOLUTION . . . . . . . . . . . . . . . . . 23 3.5 METHODS OF FINDING LAPLACE TRANSFORM . . . . . . . . . . . . . . . 23 3.6 Method of solving Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . 24 4.0 APPLICATION IN SOLVING SECOND-ORDER ODEs 27 4.1 Procedure in Solving Differential Equations using Laplace Transform . . . . . . 27 5.0 SUMMARY, CONCLUSION AND DISCUSSION 34 5.1 Discussion Of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.3 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 References 37 ABSTRACT This project work presents an approach to analytically solve second-order ordinary differential equations using the Laplace transform method. By transforming the differential equation into the Laplace domain, the problem was converted into a simpler algebraic equation. The method?ology was demonstrated through the solution of specific second-order differential equations with initial conditions. The process involves taking the Laplace transform of the differential equation, manipulating the resulting expression to isolate the transformed function, and then applying partial fraction decomposition to invert the Laplace transform and obtain the final solution in the time domain. This systematic technique provides a clear and structured method for solving a wide range of second-order differential equations, offering valuable insights into their behavior and enabling the handling of various initial conditions. CHAPTER ONE INTRODUCTION 1.1 Background Information In mathematics, the Laplace transformation deals with the transformation of one function into another function, which may or may not be in the same domain. Laplace transform. is a potent tool for transformation, especially to convert the original differential equation into an expression that can be solved Using basic algebra. The transform is named after renowned astronomer and mathematician Pierre Simon, who was a Frenchman. A logical approach for resolving linear differential equations with initial conditions is the transformation method. We can solve second-order differential equations with constant co?efficients by using the Laplace transform operator with the initial conditions Set as x = 0, differential equations must be initial value problems. problems with given Values employ the Laplace transform operator on both sides of the differential equations. Due to this, the Laplace transform of the desired solution transforms the differential problem into an algebraic equation. The Inverse Laplace transform of both sides should be taken after this algebraic problem has been solved. The outcome is the answer to the initial value issues It is vital to understand what the Laplace transform does to y ? and y ?? before starting this process. 1.2 Definition of Terms Definition 1.2.1 A differential equation is an equation involving an unknown function and its derivatives. Math?ematically defined as f f x, y(x), y? (x), · · · , y(n) (x)
= 0 where x and y(x) are the independent variables and dependent variables respectively. y ? (x), · · · y n (x) are the derivatives of the dependent variable y(x) from the first order up to the nth order. Examples d dx = 7x + 5 (1.1) 6d 3 y dx3 + (sin x) d 2 y dx2 + 7xy = 0 (1.2) ? 2 y ?t2 + 622 y 2x 2 = 0 (1.3) Types of Differential Equation 1. Ordinary Differential equation (ODE) 2. Partial Differential Equation (PDE) Definition 1.2.2 A differential equation is an Ordinary differential equation (ODE) if the Unknown functions depend on only one independent variable. Examples 1.1 and 1.2 above are ODE since the Unknown function y depends on the variable x. Definition 1.2 .3 A differential equation is a partial differential equation (Pde) If the unknown function depends on two or more independent Variables. Example 1.3 above is a PDE since y depends on both the independent variables x and t. Definition 1.2.4 Order of A Differential Equation The order of a differential equation is the order of the highest derivative appearing in the equation. It is also called the differential coefficient. Example 1. dy dx + (x 2 + 5) y = x 5 This is a first-order D.E 2. d 2 y dx2 + + x 3 + 3x
y = 9  d 2 y dx2 3 + dy dx = sin x ? ? ? This is a second order D. . E, since the highest derivative involved is a second derivative. Ordinary Differential Equation and System A normal system of first-order ODE is y ? 1 = f1 (t, y1, . . . , yn) y ? 2 = f2 (t, y1, . . . , yn) . . . y ? n = fn (t, y1, . . . , yn) where f, . . . , fn are n-given functions defied in some region D of the (n + 1)-dimentional Eu?clidean space. SECOND ORDER ODE A second order ode has the form d 2 y dt2 = f  t, y, dy dt  . where t is some given function, usually, we denote the independent variable by t since time is often the independent variable in physical problems, but sometimes we will use x instead. The equation above is said to be linear if the function t has the form f  t, y, dy dt  = g(t) ? p(t) dy dr ? q(t)y The f is linear in y and y ? · g, p and q are specified functions of the independent variable t but do not depend on y. 1.3 Statement of The Problem The problem statement for solving second-order ordinary differential equations (ODEs) using Laplace Transform involves addressing the challenges faced in transforming the given ODE into the Laplace domain and then solving it. The main issues include determining the appropriate initial conditions, ensuring the existence of the Laplace transform for the given ODE, and handling complex algebraic manipulations during the inverse Laplace transform. Additionally, the problem statement involves finding solutions that satisfy the boundary conditions and interpreting the results in the time domain. The aim is to develop a systematic approach that simplifies the process of solving second-order ODEs using Laplace Transform, making it more efficient and accurate. 1.4 Aim and Objective of The Study The aim of this work is to present the application of Laplace Transpire of Ordinary differential equations of second order. The specific objectives are: i. Study the properties of Laplace Transform ii. Apply Laplace Transform to solve some examples of second-order Ordinary differential equations. iii. Determine the uses of the Laplace transform in solving differential equations of real-life problem 1.5 Significance of The Study The study on the application of Laplace transforms in solving second-order ordinary differential equations will benefit students in the mathematics department and other researchers who may wish to conduct similar research on the above topic because will educate them on how to apply Laplace transforms in solving differential equations of second order derivatives. Finally, the study will add to the body of information and literature already present in this area of study and serve as a foundation for additional research. 1.6 Scope of Study The research is on the application of Laplace transform in solving ordinary differential equation of second-order. The study will cover and be limited on how to apply Laplace transform to ordinary differential equations of second order.