TEXAS A&M UNIVERSITYDEPARTMENT OF MATHEMATICSMATH 308-506Exam 1.A, 21 Sep 2005Name: Mark: /401. Sketch the direction field for the equationdxdt= 5x(x − 1)2(2 − x)in the region 0 ≤ t ≤ 1, 0 ≤ x ≤ 2.5. What is the t → ∞ limit of the solution satisfyingx(0) = 1.5? Can the solution satisfying x(0) = 0.5 ever grow to 1.5? Justify.(8 marks)2. Solve the IVPdxdt= x2(1 + sin(t)), x(0) = 1.(8 marks)3. Solve the IVPcos(x)dydx+ 2 sin(x)y = x cos3(x), y(0) = 2.(8 marks)4. The body of a murder victim was discovered at 6pm. Police officers measured the bodytemperature at 6.10pm and then again at 7.10pm; the temperature readings were 29oC and25oC correspondingly. The temperature of the building is maintained by an air conditioningsystem at the constant 21oC. Assuming the victim had the normal temperature of 37oC atthe time of the murder, what time did the murder happen? Use Newton’s law of coolingdTdt= k(M − T ),where T is the temperature of the body, M is the temperature of the environment andk is a proportionality coefficient. Write down the equations you are solving. (Hint: take6.10pm as t = 0).(8 marks)5. A sailboat has been running (on a straight course) under a light wind at 1 m/sec. Suddenlythe wind picks up, blowing hard enough to apply a constant force of 600 N to the boat.The only other force on the boat is water resistance that is proportional to the velocity ofthe boat with the proportionality constant b = 100 N-sec/m. If the mass of the boat is150 kg, find the velocity of the boat as a function of t. What is the limiting velocity of theboat.(8
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