This chapter introduces the Laplace transform—an integral transform often used by engineers and scientists in industry and academia to solve linear differential equations with constant coefficients, especially if those equations involve Heaviside and Dirac delta functions (see Chap. 7 ). Section 11.1 begins with the initial value problem \(\displaystyle \frac {dx}{dt} + 2x = 3e^t, \quad x(0) = 1, \qquad \hbox{(1)} \) where it is shown that the solution of (1) can be found by multiplying both sides of the differential equation by \(e^{-st}\) (without specifying a value for s) and then integrating from 0 to \(\infty \) with respect to t . This leads to the definition of the Laplace transform, which is employed in Sect. 11.2 to obtain Laplace transforms of basic functions, such as \(e^{bt}\) , \(t^n\) , \(\sin {bt}\) , and the Heaviside and Dirac functions. These transforms are among those included in a short table titled “Basic Laplace Transforms.” The examples in Sect. 11.3 illustrate how to use this table together with the linearity property of the transform to solve initial value problems like (1) without having to resort to the definition. Section 11.4 introduces the class of functions of exponential order. This sets the stage for deriving the formula for the Laplace transform of the first derivative and then shows how to use it and a version of Lerch’s theorem to solve first-order initial value problems. Piecewise continuous functions and the existence and long-term behavior of Laplace transforms are covered in Sect. 11.5. The shift (translation) properties are derived in Chap. 6 along with other important properties of the Laplace transform, such as the formulas involving derivatives and integrals of Laplace transforms. Formulas for computing transforms of periodic functions and the convolution of two functions are also derived. Examples abound illustrating how to use these properties to find the transforms of functions that are not listed in the aforementioned table of “Basic Laplace Transforms.” Although the notion of an inverse Laplace transform of a function is evident from solving the first-order initial value problems in Sect. 11.4, it is formally defined in Sect. 11.7 and illustrated with many examples. The shift properties are restated in a form more amenable for finding inverse transforms. This section includes a number of results stating the conditions under which inverse Laplace transforms of functions of the form \(\displaystyle F(s) = \frac {G(s)}{1 - ae^{-bs}}, \qquad \hbox{(2)} \) are periodic, where a, b are constants with \(a \neq 0\) and \(b>0\) . Section 11.8 is replete with examples illustrating how to use the transform formulas for first and second derivatives and the properties derived in the previous sections to solve second-order initial value problems with a variety of different forcing functions. Furthermore, it is shown in Sect. 11.9 that even a certain type of second-order differential equation with variable coefficients can be solved with Laplace transform methods.

Laplace Transforms

章节
发表日期：

2026年

查看全文

Leigh C. Becker

摘要

This chapter introduces the Laplace transform—an integral transform often used by engineers and scientists in industry and academia to solve linear differential equations with constant coefficients, especially if those equations involve Heaviside and Dirac delta functions (see Chap. 7 ). Section 11.1 begins with the initial value problem \(\displaystyle \frac {dx}{dt} + 2x = 3e^t, \quad x(0) = 1, \qquad \hbox{(1)} \) where it is shown that the solution of (1) can be found by multiplying both sides of the differential equation by \(e^{-st}\) (without specifying a value for s) and then integrating from 0 to \(\infty \) with respect to t . This leads to the definition of the Laplace transform, which is employed in Sect. 11.2 to obtain Laplace transforms of basic functions, such as \(e^{bt}\) , \(t^n\) , \(\sin {bt}\) , and the Heaviside and Dirac functions. These transforms are among those included in a short table titled “Basic Laplace Transforms.” The examples in Sect. 11.3 illustrate how to use this table together with the linearity property of the transform to solve initial value problems like (1) without having to resort to the definition. Section 11.4 introduces the class of functions of exponential order. This sets the stage for deriving the formula for the Laplace transform of the first derivative and then shows how to use it and a version of Lerch’s theorem to solve first-order initial value problems. Piecewise continuous functions and the existence and long-term behavior of Laplace transforms are covered in Sect. 11.5. The shift (translation) properties are derived in Chap. 6 along with other important properties of the Laplace transform, such as the formulas involving derivatives and integrals of Laplace transforms. Formulas for computing transforms of periodic functions and the convolution of two functions are also derived. Examples abound illustrating how to use these properties to find the transforms of functions that are not listed in the aforementioned table of “Basic Laplace Transforms.” Although the notion of an inverse Laplace transform of a function is evident from solving the first-order initial value problems in Sect. 11.4, it is formally defined in Sect. 11.7 and illustrated with many examples. The shift properties are restated in a form more amenable for finding inverse transforms. This section includes a number of results stating the conditions under which inverse Laplace transforms of functions of the form \(\displaystyle F(s) = \frac {G(s)}{1 - ae^{-bs}}, \qquad \hbox{(2)} \) are periodic, where a, b are constants with \(a \neq 0\) and \(b>0\) . Section 11.8 is replete with examples illustrating how to use the transform formulas for first and second derivatives and the properties derived in the previous sections to solve second-order initial value problems with a variety of different forcing functions. Furthermore, it is shown in Sect. 11.9 that even a certain type of second-order differential equation with variable coefficients can be solved with Laplace transform methods.