962EDB8Cd01 - x-x 2 sin y + e y ) dy + (3 y 3 e x + 2 x cos...

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M427K EXAM 1B Your name: SPRING, 2010 Dr. Schurle Your UTEID: Show all your work on these pages. Be organized and neat. Your work should be your own; there should be no talking, reading notes, checking laptops, using cellphones, . ... 1. (12 points) Draw (our version of) a direction field for the differential equation y 0 = ( y + 3) 2 ( y - 2)( y - 6) . Your y -axis must have a scale. Determine the behavior of y ( t ) as t → ∞ . If this behavior depends on the initial value y (0), describe fully this dependency. 2. (14 points) Find the general solution of each of the following differential equations. (a) y 00 + 3 y 0 - 18 y = 0 (b) 9 y 00 - 6 y 0 + 10 y = 0
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YOUR SCORE: /100 3. (12 points) Find the general solution of the following differential equation, in explicit form if possible. 2 y 0 = 4 x 3 y 3 + 3 x 2 y 3 y - 1 4. (12 points) Is the following differential equation exact? If so, solve it. If not, try multiplying by x 2 first. (9 y 2 e
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Unformatted text preview: x-x 2 sin y + e y ) dy + (3 y 3 e x + 2 x cos y-cos x ) dx = 0 5. (12 points) Find the general solution of ( t 2 + 1) y + 4 ty = sin t t 2 + 1 + t. 6. (14 points) A 1000 liter tank is full of water in which 2000 grams of dye is dissolved. A solution containing 1 gram of dye per liter runs in at 20 liters per minute, and the well-stirred solution runs out at 30 liters per minute. How much dye is in the tank after 90 minutes? 7. (12 points) Suppose y ( t ) satises y = 3 t 2 + 2 y 2 and y (1) = 2. Use Eulers Method with step size h = t = 0 . 1 to approximate y (1 . 2). DO NOT ROUND OFF! 8. (12 points) For the initial value problem y = 3 t 2 + y 3 , y (0) = 0, let ( t ) = 0 and use the method of successive approximations to calculate 1 ( t ) and 2 ( t )....
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962EDB8Cd01 - x-x 2 sin y + e y ) dy + (3 y 3 e x + 2 x cos...

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