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Unformatted text preview: Exam 1, Version A Math 20D, Lecture C, Winter 2006 6 February 2006 Name: ID #: Section Time: # Score 1a 1b 1c 2 3a 3b 3c 4a 4b Total Read all of the following information before starting the exam: Show all work, clearly and in order, if you want to get full credit. I reserve the right to take off points if I cannot see how you arrived at your answer (even if your final answer is correct). Justify your answers algebraically whenever possible to ensure full credit. When you do use your calculator, sketch all relevant graphs and explain all relevant mathematics. Circle or otherwise indicate your final answers. Please keep your written answers brief; be clear and to the point. I will take points off for rambling and for incorrect or irrelevant statements. This test has 4 problems and is worth 100 points. It is your responsibility to make sure that you have all of the pages! Good luck! 1. [ 24 points ] Determine whether each of the following equations is linear, separable, exact, or none of the above in other words, identify a solution approach for the problem. Some of these may satisfy multiple labels pick one. Do not solve the differential equation. a. [ 8 pts ] dy dx = xy 3 (1 + x 2 )- 1 / 2 Solution: We re-write the equation as y- 3 dy dx- x (1 + x 2 )- 1 / 2 = 0 . This is in separable form, so the equation is separable . Moreover, we note that M y ( x ) = N x ( y ) = 0 , so it is also exact . The equation is not linear. b. [ 8 pts ] dy dx = 3 x 2- 2 xy + 2 x 2- 6 y 2- 3 Solution: We multiply through by the denominator of the right-hand side to get ( x 2- 6 y 2- 3) dy dx + (2 xy- 3 x 2- 2) = 0 This is in exact form, but first we must check that it is exact. Differentiating the first term byThis is in exact form, but first we must check that it is exact....
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- Spring '06