Remember to review all aspects of techniques of integration from your 69.104 course. For a quick review look at the 69.104 course outline from the interactive Weekly Quiz 6 up to the end. Separable equations are used in population biology and many other areas. Newton's Law of Cooling appplies to most solids cooling down in a thermal environment.
For another application (which involves numerical calculation as well) to flows from damaged oil tanks check out this link.
Remember to review all aspects of techniques of integration by parts this week; they will come in handy later. As pointed out in class, we need this material in order to work on exact differential equations. The business on partial derivatives is really important so please look and work out as many of the homework problems as possible.
Test coming up next week!
By then you will have looked at some 100 problems, Tutorials and Homework, included. I will choose 4-6 questions out of all these problems for your one-hour Test. It will be held during your normal tutorial hour, and, you must write your test during your regularly scheduled hour. Answers to Tutorial 2 will be up on this site by Thursday, while answers to your homework problems (the odd numbers) are (as usual) in the back of your text. For this reason, a Sample Test will not be necessary.Test this week! See the notes above for info.
Old Test 1 Solutions:
Questions 1a and 1b
Questions 2, 3, and 4. In question No. 3, the coefficient of y' should be x2+2xy and not the term listed.
Shortcut in Integration by Parts, I
NOTE: You don't need to look at this section if you have my {\it Calculus} book!Shortcut in Integration by Parts, II
Shortcut in Integration by Parts, III
Example using Variation of Parameters
Example using Variation of Parameters, continued
Another example using Variation of Parameters
The Cauchy-Euler equation is covered in your text and these notes will act as supplement to this topic. You may use Chapter 17 in Schaum's Outline Series in Differential Equations, by Frank Ayres Jr., (McGraw-Hill) for additional problem sets.
Old Test 2 Solutions:
Questions 1 and 2
Questions 3 and 4
Material covering the test includes all homework problems from Chapter 10, specifically, Sequences and Series and the Integral Test for series. Although remainders were covered in class, I forgot to assign problems associated with them... so you're all off the hook!
Old Test 3, Solutions:
Questions 1 and 2
Questions 3 and 4
Material covering Test 4 includes all homework problems (including Remainders), alternating series and their remainders, the Ratio and Root Tests, Power series and basic Maclaurin series and Taylor Series.
Page 1
Old Test 4, Solutions:
Questions 1 and 2
Questions 3 and 4
