FinalExamReviewQuestions - ME 163 SPRING 2000 FINAL EXAM...

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ME 163 SPRING 2000 FINAL EXAM REVIEW WEB SITE Any changes in office hours or any corrections, comments or hints on the review problems will be posted on the web site. Please check it frequently between now and the exam. The address of the web site is . TIME AND PLACE OF EXAM The exam will be on Saturday May 6, 4:00 - 7:00 PM in Morey 321 (the regular MWF classroom). MATERIAL COVERED BY EXAM The exam will cover everything in the course. It will cover all of the material of homework assignments #1-11, and the project. It will cover the following sections in the text: 1.1-1.3, 1.5, 2.1-2.4, 2.6, 3.2-3.4, 4.1-4.9, 4.11-4.12, 5.1-5.2, 12.2-12.4, 12.6. The exam will be open book and notes - any reference material may be used, but you may not exchange reference material during the exam. A calculator will be useful for some of the problems. OFFICE HOURS BEFORE THE EXAM My office hour schedule is given below. Any changes will be posted on the web site. You are welcome to come at other times, but you might want to call or E-mail first to make sure that I will be there (x54078; [email protected]). THUR APRIL 27 4:00 - 6:00 PM FRI APRIL 28 4:00 - 6:00 PM MON MAY 1 4:00 - 6:00 PM TUE MAY 2 4:00 - 6:00 PM WED MAY 3 4:00 - 6:00 PM THUR MAY 4 4:00 - 6:00 PM FRI MAY 5 4:00 - 6:00 PM PRACTICE FINAL EXAM The last pages of this handout are the final exam and solutions from 1999. After you have finished your basic review, try to work this exam in three hours as practice and use the results to identify any areas that need further study. Note that the exam last year did not include material on
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ME 163 FINAL EXAM REVIEW PAGE 2 conservative periodic solutions and limit cycles (material covered in the last four lectures this year). The review problems below do have problems on these topics. SUGGESTED REVIEW PROBLEMS You should review the homework problems and the examples given in class. Also try to work all of the problems given here. Any corrections, additional hints or comments on these problems will be posted on the web site. Most of these problems are from Fundamentals of Differential Equations and Boundary Value Problems , by R. Kent Nagle and Edward B. Saff. FIRST ORDER EQUATIONS Basic Concepts, Solution Curves and Direction Fields (1) Show that y x 2 3 0 + = is an implicit solution of dy dx y / /( ). = − 1 2 (2) Consider the initial value problem y dy dx x y x y = = 4 0 0 0 , ( ) . What does the basic existence and uniqueness theorem tell you about the solution of this problem? What if y 0 = 0 ? (3) Sketch the direction field for the equation dy dx y = 1 4 . From your sketch, what can you say about the solutions of this equation as x → ∞ ? Separable Equations (4) Determine whether the following equation is separable: dy dx ye x x y = + + 2 2 . (5) Solve dy dx x y y = = 1 1 1 2 4 , ( ) . with (Answer: y x x x ( ) . = ( ) 5 5 3 3 7 3 1 5 ) (6) Solve the initial value problem dy dx y x y = + = ( )tan( ) , ( ) . 1 0 3 2 (Answer: y x x ( ) tan log(cos ) . = [ ] π 3 ) (7) A bank account pays interest at a rate r , compounded continuously. The initial balance is zero, but principal is added at a rate Γ , also continuously. If Γ
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