5: Laplace Transforms
- Page ID
- 89124
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"We could, of course, use any notation we want; do not laugh at notations; invent them, they are powerful. In fact, mathematics is, to a large extent, invention of better notations." - Richard P. Feynman (1918-1988)
- 5.1: The Laplace Transform
- Up to this point we have only explored Fourier exponential transforms as one type of integral transform. The Fourier transform is useful on infinite domains. However, students are often introduced to another integral transform, called the Laplace transform, in their introductory differential equations class. These transforms are defined over semi-infinite domains and are useful for solving initial value problems for ordinary differential equations.
- 5.2: Properties and Examples of Laplace Transforms
- IT IS TYPICAL THAT ONE MAKES USE of Laplace transforms by referring to a Table of transform pairs. A sample of such pairs is given in Table 5.2.1 . Combining some of these simple Laplace transforms with the properties of the Laplace transform, as shown in Table 5.2.2 , we can deal with many applications of the Laplace transform.
- 5.3: Solution of ODEs Using Laplace Transforms
- ONE OF THE TYPICAL APPLICATIONS OF LAPLACE TRANSFORMS is the solution of nonhomogeneous linear constant coefficient differential equations. In the following examples we will show how this works.
- 5.4: Step and Impulse Functions
- OFTEN, THE INITIAL VALUE PROBLEMS THAT ONE FACES in differential equations courses can be solved using either the Method of Undetermined Coefficients or the Method of Variation of Parameters. However, using the latter can be messy and involves some skill with integration. Many circuit designs can be modeled with systems of differential equations using Kirchoff’s Rules.
- 5.5: The Convolution Theorem
- Finally, we consider the convolution of two functions. Often, we are faced with having the product of two Laplace transforms that we know and we seek the inverse transform of the product.
- 5.6: Systems of ODEs
- LAPACE TRANSFORMS ARE ALSO USEFUL for solving systems of differential equations. We will study linear systems of differential equation in Chapter 6. For now, we will just look at simple examples of the application of Laplace transforms.