Chapter 8 - Laplace transform methods

Abstract

In previous chapters we have investigated solving the nth-order linear equation(8.1)an(t)y(n)+an−1(t)y(n−1)+⋯+a2(t)y″+a1(t)y′+a0(t)y=f(t) for y. We have seen that if the coefficients ai(t) of (8.1) are numbers, we can find a general solution of the equation by first solving the characteristic equation of the corresponding homogeneous equation, forming a general solution of the corresponding homogeneous equation, and then finding a particular solution to the nonhomogeneous equation. If the coefficients ai(t) are not constants, we have learned that solving (8.1) may be substantially more difficult than in other cases, such as when the functions ai(t) are constants. For example, when Eq. (8.1) is a Cauchy–Euler equation, techniques used to solve the case when (8.1) has constant coefficients can be used to solve the equation. In other situations, we might be able to use a series to find a solution of the equation. Regardless, in all these situations the forcing function f(t) has been a smooth function. If f(t) is not a smooth function, such as when f(t) is a piecewise defined or a periodic function with discontinuities, solving Eq. (8.1) is substantially more difficult to solve using the techniques that we have discussed. In this chapter, we discuss a technique that transforms Eq. (8.1) or systems of linear differential equations into an algebraic equation or equations that can sometimes be solved so that a solution to the differential equation or system of differential equations can be obtained.