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M251Hex1_mockup(sp09)

# M251Hex1_mockup(sp09) - same rate Let Q t be the amount of...

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Math 251H First Midterm Exam 75 minutes February 20, 2009 SAMPLE EXAM. 1

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1. (10 points) Find the general solution of the ﬁrst order diﬀerential equa- tion, t 2 y 0 - 2 ty = 3 .
2. (15 points) For the autonomous equation y 0 = y ( y - 2)( y + 4) . a. (5 points) Find all equilibrium solutions. b. (5 points) Determine which of them are stable. Justify your answer. c. (5 points) Determine the behavior of solution y ( t ), which satisﬁes the initial value y (1) = 1, when t + .

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3. (15 points) Consider the initial value problem: 6 x 2 - 2 xy + e x + y + ( e x + y - x 2 ) dy dx = 0 ,y (1) = - 1 . (a) (5 points) Verify that the equation is exact. (b) (10 points) Solve this initial value problem. You may leave your answer in implicit form.
4. (15 points) A tank initially contains 100 liters of pure water. A mixture containing a concentration of 5 grams/liter of salt enters the tank at a rate of 2 liters/min. and the well-stirred mixture leaves the tank at the

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Unformatted text preview: same rate. Let Q ( t ) be the amount of salt in the tank. Formulate and state an initial value problem satisﬁed by Q modeling this process. Make sure you write down both an equation and an initial condition that Q ( t ) must satisfy. 5. (15 points) Find the solution of the following initial value problem using the method of undetermined coeﬃcients: y 00-4 y + 3 y = t 2 + te 3 t , y (0) = 1 , y (0) = 0 . 6. (10 points) Use the method of reduction of order to ﬁnd a second solution of the diﬀerential equation: t 2 y 00 + 2 ty-6 y = 0 , t > , knowing that y 1 ( t ) = t 2 is a solution. What is the general solution of the above equation? 7. (10 points) Find α so that the solution to the initial value problem y 00 + 3 y-4 y = 0 ,y (0) = α,y (0) = 1 , converges to 0 as t → + ∞ ....
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M251Hex1_mockup(sp09) - same rate Let Q t be the amount of...

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