Murdoch University Unit Notes
Chapter 4
Laplace transforms 4.1 Introductory remarks The Laplace transform is a powerful tool for analysing and solving a number of mathematical problems, particularly differential equations. In this chapter, we will learn about the Laplace transform and how to apply it to finding the solution to initial value problems.
4.2 Definition of the Laplace transform Laplace transforms involve ‘improper integrals’, that is integrals over an infinite range. These integrals need to be handled slightly differently than normal and you should understand how to deal with them. Read section 6.1 of Boyce and DiPrima and do problems 1, 3, 5, 6, 15, 17, 21 and 23 on page 298.
4.3 Solution of initial value problems (IVP) In this section we discover the real power of the Laplace transform as a tool for solving IVPs. Read section 6.2 of Boyce and DiPrima. The crucial results are theorems 6.2.1 and 6.2.2 where it is shown that the Laplace transform can turn a differential equation into an algebraic equation. The rest of the section deals with the mechanics of solving IVPs. You might like to mark (or copy) the table on page 304—you will find this useful when doing problems later. As this is the most important part of studying Laplace transforms, you should do lots of exercises here! Do problems 1, 3, 5, 7 and 9 on page 307, and remember about partial fractions. Also do the odd numbered problems from 11 to 25. To get a better understanding of the properties of the Laplace transform, do problems 28, 29 and 32.
52
Laplace transforms
4.4 The step function One of the really nice properties of the Laplace transform is that it automatically handles ‘difficult’ functions, that is functions with discontinuities. The simplest example of this is the step function. Read section 6.3 of Boyce and DiPrima and make sure you understand theorem 6.3.1 on page 311. This theorem is sometimes called the ‘shift theorem’ (for reasons that should be clear when you have understood the theorem). Theorem 6.3.2 is similar and also important. To make sure you understand this section, do problems 1, 3, 5, 7, 9, 11, 13, 19, 20, 22 and 25 on pages 314 and 315.
4.5 ODEs with discontinuous forcing functions By combining the method from section 6.2 of Boyce and DiPrima with the transforms of step functions in section 6.3, we can solve IVPs involving forcing functions with step discontinuities. Read section 6.4 and do problems 1, 3, 5, 7 and 9 on pages 321 and 322. Notice that the shift theorem is used at the beginning of the procedure to find the Laplace transform of the forcing function and again at the end to find the inverse Laplace transform giving the solution of the IVP.
Laplace transforms 4.1 Introductory remarks The Laplace transform is a powerful tool for analysing and solving a number of mathematical problems, particularly differential equations. In this chapter, we will learn about the Laplace transform and how to apply it to finding the solution to initial value problems.
4.2 Definition of the Laplace transform Laplace transforms involve ‘improper integrals’, that is integrals over an infinite range. These integrals need to be handled slightly differently than normal and you should understand how to deal with them. Read section 6.1 of Boyce and DiPrima and do problems 1, 3, 5, 6, 15, 17, 21 and 23 on page 298.
4.3 Solution of initial value problems (IVP) In this section we discover the real power of the Laplace transform as a tool for solving IVPs. Read section 6.2 of Boyce and DiPrima. The crucial results are theorems 6.2.1 and 6.2.2 where it is shown that the Laplace transform can turn a differential equation into an algebraic equation. The rest of the section deals with the mechanics of solving IVPs. You might like to mark (or copy) the table on page 304—you will find this useful when doing problems later. As this is the most important part of studying Laplace transforms, you should do lots of exercises here! Do problems 1, 3, 5, 7 and 9 on page 307, and remember about partial fractions. Also do the odd numbered problems from 11 to 25. To get a better understanding of the properties of the Laplace transform, do problems 28, 29 and 32.
52
Laplace transforms
4.4 The step function One of the really nice properties of the Laplace transform is that it automatically handles ‘difficult’ functions, that is functions with discontinuities. The simplest example of this is the step function. Read section 6.3 of Boyce and DiPrima and make sure you understand theorem 6.3.1 on page 311. This theorem is sometimes called the ‘shift theorem’ (for reasons that should be clear when you have understood the theorem). Theorem 6.3.2 is similar and also important. To make sure you understand this section, do problems 1, 3, 5, 7, 9, 11, 13, 19, 20, 22 and 25 on pages 314 and 315.
4.5 ODEs with discontinuous forcing functions By combining the method from section 6.2 of Boyce and DiPrima with the transforms of step functions in section 6.3, we can solve IVPs involving forcing functions with step discontinuities. Read section 6.4 and do problems 1, 3, 5, 7 and 9 on pages 321 and 322. Notice that the shift theorem is used at the beginning of the procedure to find the Laplace transform of the forcing function and again at the end to find the inverse Laplace transform giving the solution of the IVP.
