Page 1 of 5Student Name:Student Net ID:MATH 286 SECTION G1 – Introduction to Differential Equations PlusMIDTERM EXAMINATIONSeptember 20, 2012INSTRUCTOR: M. BRANNANINSTRUCTIONS• This exam 50 minutes long. No personal aids or calculators are permitted.• Answer all questions in the space provided. If you require more space to write youranswer, you may continue on the back of the page. There is a blank page at the endof the exam for rough work.• EXPLAIN YOUR WORK! Little or no points will be given for a correct answer withno explanation of how you got it. If you use a theorem to answer a question, indicatewhich theorem you are using, and explain why the hypotheses of the theorem are valid.• GOOD LUCK!PLEASE NOTE: “Proctors are unable to respond to queries about the interpretation ofexam questions. Do your best to answer exam questions as written.”Question: 1 2 3 4 5 TotalPoints: 12 8 10 10 10 50Score:Student Net ID: MATH 286 G1 Page 2 of 51. (a) (10 points) Solve the initial value problemxdydx− y = x cos2x + yx; y(e) = 0.(b) (2 points) Is the solution you found unique?2. (8 points) Find the general solution to the ODE1xdydx+ 2ex2y − ex2= 0 (x 6= 0).Student Net ID: MATH 286 G1 Page 3 of 53. (10 points) For what values of a and b is the equation(ye2xy+ x(1 + x) + y2)dx + (axy + bxe2xy)dy = 0exact? Find an implicit solution to this equation using these values of a and b.4. (10 points) Find the general solution to the ODEdydt=ty + te−y− ye−t− e−y−t1 − e−yin implicit form.Student Net ID: MATH 286 G1 Page 4 of 55. (10 points) A 1 kg rocket is launched from the ground at an angle of 0 < θ <π2radi-ans relative to the horizon, with an initial air speed of 10 m/s. Suppose that the dragforce on the rocket due to wind resistance is proportional to the rocket’s velocity. If theconstant of proportionality relating the drag to the velocity is k = 1 N· s/m and theacceleration due to gravity is −10 m/s2, what must the angle θ be so that the rocketreturns to the ground after 1 second of flight?(Note: Since you don’t have a calculator, an algebraic expression for θ is perfectlyfine.)Student Net ID: MATH 286 G1 Page 5 of 5(Extra work
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