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E48F5E07d01

# E48F5E07d01 - e y 2 y cos x-cos y dy(9 x 2 e y-y 2 sin x e...

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M427K EXAM 1A 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 + 2)( y - 3)( y - 5) 2 . 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 + 5 y 0 - 14 y = 0 (b) 4 y 00 - 4 y 0 + 17 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. y 0 = 3 x 2 y 3 + 2 xy 3 y - 2 4. (12 points) Is the following diﬀerential equation exact? If so, solve it. If not, try multiplying by x 2 ﬁrst. (3 x 3

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Unformatted text preview: e y + 2 y cos x-cos y ) dy + (9 x 2 e y-y 2 sin x + e x ) dx = 0 5. (12 points) Find the general solution of ( t 2 + 1) y + 4 ty = e t/ 2 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 30 liters per minute, and the well-stirred solution runs out at 40 liters per minute. How much dye is in the tank after 90 minutes? 7. (12 points) Suppose y ( t ) satisﬁes y = 2 t 2 + 3 y 3 and y (1) = 1. Use Euler’s Method with step size h = Δ t = 0 . 1 to approximate y (1 . 2). DO NOT ROUND OFF! 8. (12 points) For the initial value problem y = 2 t + y 3 , y (0) = 0, let φ ( t ) = 0 and use the method of successive approximations to calculate φ 1 ( t ) and φ 2 ( t )....
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E48F5E07d01 - e y 2 y cos x-cos y dy(9 x 2 e y-y 2 sin x e...

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