1. Test Corrections You may correct your test. You will get back 1/3 of the points you lost if you submit correct answers. This work is to be done on your own (or in consultation with me only). Skidmore Honor Code! Corrections should be done on separate sheets, NOT on the original test. Hand both things in, NOT stapled together. Due on Monday (10/21) at class time.

2. Clicker Question 1 What is the general solution to the DE y ' = x sin(x 2 ) ? – A. y = x 2 cos(x 2 ) + C – B. y = -x 2 cos(x 2 ) + C – C. y = (1/2) cos(x 2 ) + C – D. y = (-1/2) cos(x 2 ) + C – E. y = - cos(x 2 ) + C

3. Clicker Question 2 Consider the DE y ' = 1/y and the function y =  (2x + 3). Then: – A. y is not a solution of the DE. – B. y is the unique solution of the DE. – C. y is one of a family of solutions of the DE.

4. Direction Fields and Euler’s Method (10/16/13) Old story: If we can’t get exact answers, then try using a limited amount of data to get approximate answers. Here, if we can’t get an explicit solution to a given DE, perhaps we can draw a picture of one or more approximate solutions.

5. Direction Fields Given a DE, a direction field for it is a picture of a grid of (x, y) points, each with an arrow attached which shows the direction a solution would be moving. These are laborious to produce without technology. So let’s use technology! Try dy/dx = x. Try dy/dx = y.

6. Euler’s Method This uses the same idea, but we start at one point and continue building from that point. Start at (x 0, y 0 ) and compute dy/dx (i.e., the slope) at that point. Now move to the right a small distance  x and move up (or down) by the computed slope times  x to a new point (x 1, y 1 ). Now repeat. The smaller  x is, the more data we will use over a given interval, and so the more accurate the resulting “curve” will be.

7. An Example of Euler’s Method Use Euler’s Method to estimate y(4) using 4 steps given the DE y = x 2 / y and the starting point y(0) = 1. Can you see how to solve this DE explicitly? If so, what is the exact answer for y(1)?

8. Assignment for Friday Read Section 9.2. In that section, please do Exercises 1, 3, 4, 5, 6, and 23. (This last, even with only 6 data points, is a pain, but it’s important to work through one of these examples by hand to make sure you understand the idea.) Work on test corrections.