finexS11 - Problems from Final exam, Spring 2011. 1. Solve...

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Unformatted text preview: Problems from Final exam, Spring 2011. 1. Solve the differential equations: a) (10 points) y = xy + y (Find the general solution) b) (10 points) x2 y = y ln y − y (Find the general solution) c) (10 points) y = 2 sin x y with initial condition y (π/2) = −4. 2. Solve the differential equation (find the general solution): a) (5 points) y + y − 20y = 0. b) (5 points) y − 8y + 12y = 0. 3. (10 points) Solve the following initial value problem: y + 6y + 9y = 0; y (1) = 5, y (1) = −2. 4. (8 points) A tank holds 60 pounds of salt dissolved in 120 gallons of water. At time t = 0 brine containing 3 pounds of salt per gallon begins to flow into the tank at the rate of 2 gallons per minute and continues to do so. Beginning at the same time the perfectly mixed solution flows out of a tap at the bottom of the tank at the same rate. How much salt is dissolved in the tank after 60 minutes? 5. (7 points) A cup of coffee has temperature 90 degrees Centigrade in a room kept at 20 degrees Centigrade. After half an hour the coffee cools to 80 degrees Centigrade. What is the temperature after another half hour? 6-9. Test for convergence (no credit for answer without a correct complete justification). If you use a test, you must show how it works, i.e. you can’t just say :”comparison test” or ”integral test”, you have to make the comparison or compute the appropriate integral. 6) (5 points) ∞ 1 + sin4 4n . n4 n=1 7) (5 points) ∞ 1 √ . n=2 n ln n 1 8) (5 points) ∞ sin3 4n . n4 n=1 9) (5 points) ∞ ln n . 2 n=1 n 10. Consider the power series ∞ (−1)n−1 n=1 (3x)n . n a) (4 points) Find the radius of convergence. b) (4 points) Find the interval of convergence. 11. (7 points) a) Find the Taylor series for f (x) = 1 x2 at a = 1. b) What is the radius of convergence? 2 ...
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finexS11 - Problems from Final exam, Spring 2011. 1. Solve...

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