Chapter 9 - Applications of the Laplace Transform

Abstract

Laplace transforms are useful in solving the spring-mass systems. Although the method of Laplace transforms can be used to solve problems of higher order equations, this method is most useful in alleviating the difficulties associated with problems of the type that involves piecewise-defined forcing functions. This chapter reviews the use of Laplace transforms to solve the second order initial value problem that models the motion of a mass attached to the end of a spring: mx”+cx’+kx = f(t) x(0) = x0, x’(0) = v0, where m represents the mass, c the damping coefficient, and k the spring constant determined by Hooke's law. The chapter illustrates how the method of Laplace transforms is used to solve this type of initial value problem in which the forcing function is continuous with the help of some examples. Laplace transforms can be used to solve the L-R-C circuits problems as well. This method is most useful when the supplied voltage v(t) is piecewise-defined. Laplace transforms can be used to solve the population problems. In this case, however, the chapter focuses on the problems that include a nonhomogeneous forcing function. Laplace transforms are especially useful when dealing with piece wise-defined forcing functions, but they are useful in many other cases as well. The chapter reviews a problem involving a continuous forcing function used to describe the presence of immigration or emigration. In many cases, it is required to determine the inverse Laplace transform of a product of two functions. Just as in integral calculus when the integral of the product of two functions did not produce the product of the integrals, the inverse Laplace transform of the product also does not yield the product of the inverse Laplace transforms. Therefore, the Convolution theorem is needed.