# 8: Laplace Transforms

- Page ID
- 9438

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IN THIS CHAPTER we study the method of Laplace transforms, which illustrates one of the basic problem solving techniques in mathematics: transform a difﬁcult problem into an easier one, solve the latter, and then use its solution to obtain a solution of the original problem. The method discussed here transforms an initial value problem for a constant coefﬁcient equation into an algebraic equation whose solution can then be used to solve the initial value problem. In some cases this method is merely an alternative procedure for solvingproblems that can be solved equally well by methods that we considered previously; however, in other cases the method of Laplace transforms is more efﬁcient than the methods previously discussed. This is especially true in physical problems dealing with discontinuous forcing functions.

- 8.1: Introduction to the Laplace Transform
- This section deﬁnes the Laplace transform and develops its properties.

- 8.2: The Inverse Laplace Transform
- This section deals with the problem of ﬁnding a function that has a given Laplace transform.

- 8.3: Solution of Initial Value Problems
- This section applies the Laplace transform to solve initial value problems for constant coefﬁcient second order differential equations on (0,∞).

- 8.4: The Unit Step Function
- In this section we’ll develop procedures for using the table of Laplace transforms to find Laplace transforms of piecewise continuous functions, and to find the piecewise continuous inverses of Laplace transforms. This section also introduces the unit step function.

- 8.5: Constant Coefficient Equations with Piecewise Continuous Forcing Functions
- This section uses the unit step function to solve constant coefﬁcient equations with piecewise continuous forcing functions.

- 8.6: Convolution
- This section deals with the convolution theorem, an important theoretical property of the Laplace transform.

- 8.7: Constant Coefficient Equations with Impulses
- This section introduces the idea of impulsive force, and treats constant coefﬁcient equations with impulsive forcing functions. We we consider initial value problems where the forcing function represents a force that is very large for a short time and zero otherwise. Impulsive forces occur when two objects collide. Since it is not feasible to represent such forces as continuous or piecewise continuous functions, we need a different mathematical model to deal with them.

- 8.8: A Brief Table of Laplace Transforms
- This section is a brief table of Laplace transforms.