Differential Equations and Linear Algebra, 4th Edition

10 Laplace Transform Methods

10.1 Laplace Transforms and Inverse Transforms

In Chapter 5 we saw that linear differential equations with constant coefficients have numerous applications and can be solved systematically. There are common situations, however, in which the alternative methods of this chapter are preferable. For example, recall the differential equations

$$m{x}^{″}+c{x}^{\text{′}}+kx=F(t)\text{and}L{I}^{″}+R{I}^{\text{′}}+\frac{1}{C}I=E^{\text{′}}(t)$$

corresponding to a mass–spring–dashpot system and a series RLC circuit, respectively. It often happens in practice that the forcing term, F(t) or $E^{\text{′}}(t)\text{,}$ has discontinuities—for example, when the voltage supplied to an electrical circuit is turned off and on periodically. In this case the methods of Chapter 5 can be quite awkward, and the Laplace ...