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.
Group 11 will present "Cleaning up the Great Lakes", pages 317-318.
Resolve questions about the homework.
Begin reviewing for the final exam. Since next class is our last meeting, there is no homework to be turned in.
Group 4: complete your project on "Duhamel's formulas", pages 425-426, to present in class on Thursday, April 30.