Practice_MT_1_Math_33B_W09

Practice_MT_1_Math_33B_W09 - Mathematics 33B - Practice...

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Unformatted text preview: Mathematics 33B - Practice Midterm 1 Official exam to be administered January 30th, 11:00am NAME (please print legibly): Your University ID Number: Your Discussion Section and TA: Signature: Calculators, notes and books may not be used in this examination. You may not receive full credit for a correct answer if insufficient work is shown. QUESTION VALUE SCORE 1 20 2 20 3 20 4 20 5 20 TOTAL 100 1 1. (20 points) Determine which equilibrium solutions of y = y ( y- 1)( y + 2) are asymp- totically stable and which are unstable. Make a rough sketch showing the behavior of the solutions. Make sure to include at least one curve to represent each type of solution. 2. (20 points) Suppose you are given a sample of radioactive material that has a half-life of 100 ln 2 years. a) How many years will it take so that only 1 4 th of the original material remains? b) If you continuously add material at a constant rate of 10 grams per year, how much material will you have after 10 years if you start with 200 grams? 3. (20 points) a) State carefully the uniqueness theorem for the differential equation dy dx = f ( x,y ). b) Find two solutions to the initial value problem dy dx = y 4 5 y (0) = 0 . c) The above initial value problem has more than one solution. Why does this not contradict the uniqueness theorem you stated above? 4. (20 points) Is the equation ( y 2- xy ) dx + x 2 dy = 0 exact? Use any method you know to solve the equation. 2 5. (20 points) True/False Questions: 1) Suppose that y = f ( x,y ) and, for all ( x,y ), f is continuous and differentiable in y . If two curves z = z ( x ) and w = w ( x ) are solutions to the above differential equation and...
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Practice_MT_1_Math_33B_W09 - Mathematics 33B - Practice...

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