test2prf3-f8-1 - MATH 2403F3-F8 Practice Problems for...

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MATH 2403F3-F8 Practice Problems for Midterm 2 Fall 2009 TIME: October 1 (Thursday), 2009, 12:05-13:05 (the length of the exam is 60 min) PLACE: Howey Physics L1 (where we meet on Tuesdays & Thursdays) COVERAGE: Lectures 09/08-09/24 EXAM POLICY: No calculator. No books. No notes, except for one formula sheet of letter size handwritten by yourself. Turn in this formula sheet with your test. The problems below are only meant for your practice. The real problems on the exam will not be the same. The amount of questions will also not be the same. Good luck! [1] For each of the following equations, find general solutions; solve the initial value problem with initial condition x 1 (0) = - 1 , x 2 (0) = 2; sketch the phase portrait, identify the type of each equilibrium, and determine the stability of each equilibrium. (a) dx 1 dt = - 2 x 1 - 13 x 2 , dx 2 dt = x 1 +4 x 2 (b) dx 1 dt = x 1 + x 2 , dx 2 dt = - 2 x 1 - x 2 (c) dx 1 dt = 3 2 x 1 - 4 x 2 , dx 2 dt = 5 x 1 - 5 2 x 2 (d) dx 1 dt = x 1 +4 x 2 , dx 2 dt = - x 1 +5 x 2 (e) dx 1 dt = - 5 x 1 - x 2 , dx 2 dt = 4 x 1 - x 2 (f) dx 1 dt = - 2 x 1 , dx 2 dt = - 2 x 2 [2] Find all equilibria, identify the type of each equilibrium, and determine the stability of each equilibrium: (a) dx 1 dt = (4 - x 1 - x 2 ) x 1 , dx 2 dt = ( - 2 + x 1 ) x 2 (b) dx 1 dt = - x 2 1 + x 2 2 + 3 , dx 2 dt = - x 1 + 2 x 2 [3] For each of the following equations, find general solutions; solve the initial value problem with initial condition y (0) = - 1 , y (0) = 2;
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