Read sections 7.7-7.9, pages 402-420, in Nagle & Saff.
Work the following problems from Nagle & Saff.
Section 7.5, page 388, exercises 25 and 35.
Section 7.6, page 399, exercises 7 and 13.
Section 7.7, pages 409-410, exercises 5, 13, 15, 23.
Section 7.8, page 417, exercises 7 and 13.
Section 7.9, page 420, exercise 1.
Explore further applications of the Laplace transform.
As the end of the semester nears, it is appropriate to look back at what you have learned and to integrate your knowledge. The homework today has two goals: to revisit some previous topics, and to see what new insights are available by using Laplace transforms.
Revisit exercise 32, section 2.3, page 54 of Nagle & Saff. You previously worked this exercise by solving the differential equation on two abutting intervals and choosing the integration constants to patch the solutions together at the common endpoint. Now solve the exercise by using the method of Laplace transforms.
Work exercise 2 in the Chapter 2 review problems, page 79 of Nagle & Saff, two ways: first by treating it as a first-order linear equation (use an integrating factor); and second by using the method of Laplace transforms.
Are your two answers compatible with each other? Which method is simpler conceptually? Which method is easier to implement?
Revisit exercise 37 in the Chapter 4 review problems, page 229 of Nagle & Saff. You previously solved this exercise by using the method of reduction of order: one solution f was given to you, deus ex machina, and you found a second solution by trying y=v(x)f(x) and solving an auxiliary differential equation for the unknown function v.
Show that you can solve the original differential equation from scratch, without being given a starting solution. Namely, Laplace transform the equation by using property 7 in Table 7.2 on page 368. You will get a first-order differential equation for Y(s), which you can solve via an integrating factor. Then take the inverse Laplace transform to get the solution y(x) to the original problem. Your final answer should be compatible with the one in the back of the book.
Groups 1, 2, 4, 8, and 11: work on your projects.
Group 1 will present "Linearization of Nonlinear Problems", pages 232-233, on Thursday, April 23.
Group 2 will present "Convolution Method", page 231, on Thursday, April 23.
Group 8 will present "Designing a Landing System for Interplanetary Travel", pages 311-312, on Tuesday, April 28.
Group 11 will present "Cleaning up the Great Lakes", pages 317-318, on Tuesday, April 28.
Group 4 will present "Duhamel's formulas", pages 425-426, on Thursday, April 30.