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 and 2: complete your projects, which you will present today.
Group 1 will present "Linearization of Nonlinear Problems", and Group 2 will present "Convolution Method".
We will discuss autonomous systems of differential equations.
The goal of this homework is to re-integrate your knowledge of systems of differential equations. We have seen different ways of solving linear systems of differential equations: for example, by eliminating variables or by using Laplace transforms. On the other hand, non-linear systems generally require different techniques.
Read section 5.7, pages 287-297, in Nagle & Saff.
Solve exercise 14, section 5.7, page 297, in two ways.
Follow the method of Example 3, page 290: use the chain rule to write an equation for dy/dx, and then solve this first-order differential equation for y(x). You will want to refer to the discussion on pages 74-75 to complete the solution.
You may use Maple to assist in the calculations. Also use Maple's implicitplot command to plot a solution curve passing through the origin of the x-y plane.
Since this autonomous system is linear, you can solve directly for x(t) and y(t). Do this by taking the Laplace transform of the pair of differential equations, solving the resulting pair of equations for X(s) and Y(s), and taking the inverse Laplace transform to get x(t) and y(t).
You may use Maple to assist in the calculations. Also use Maple's plot command to plot a trajectory passing through the origin of the x-y plane. (Substitute the initial conditions x(0)=0 and y(0)=0 and try plot([x(t), y(t), t=-1..1]);.)
Your two curves should agree. Do they?
As an application of linear systems of differential equations, set up and solve exercise 11, section 5.5, page 275. (This is similar to Example 2 on page 271.) See if you can solve both by the elimination method and by the method of Laplace transforms.
You may use Maple to assist with the calculations.
Check that your answer is compatible with the one in the back of the book.
Groups 4, 8, and 11: continue to work on your projects.
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.