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m206_reviewtest1_sp09

# m206_reviewtest1_sp09 - Spring 2009 Monika Brannick...

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Unformatted text preview: Spring 2009 Monika Brannick Diﬀerential Equations - Math 206 Review Test 1 Note: This is not a sample test. The following are just examples for you to review. 1. Give the order of each of the following diﬀerential equations , and state whether the equation is linear or nonlinear. (a) (1 − x)y − 4xy + 5y = cos x (b) (1 − y )y + 2y = ex (c) (sin θ)y − (cos θ)y = 2 2. Verify that the function y = xex is a solution of the diﬀerential equation y − 2y + y = 0. 3. Determine whether the diﬀerential equation (3xy + y 2 ) + (x2 + xy )y = 0 is exact. If it is not, ﬁnd an appropriate integrating factor to make the equation exact. 4. Find the general solution for the diﬀerential equation 2y 1 dy − 2 = x cos x, x > 0. x dx x dy = y 2 t 1 + 3t2 . dt (b) Find the solution which satisﬁes y (0) = −2. 5. (a) Solve dy 6. Consider the diﬀerential equation = y (y − 1)(y − 2), y > 0. Determine all critical points dx and classify them as asymptotically stable, unstable, or semi stable. dy 7. The diﬀerential equation (6x + 1)y 2 + 3x2 + 2y 3 = 0 can be solved using two diﬀerent techdx niques. (a) Determine which techniques can be used to solve the equation. (b) Solve the equation. (The choice of technique is yours!) The following question will only be covered on the test if we get to the material next week. 8. A tank contains 200 l of water in which 10 grams of salt are dissolved. Brine containing 2 grams per liter of salt is poured into the tank at the rate of 4 liters per minute, and the well-mixed solution ﬂows out at the same rate from a spigot at the bottom of the tank. (a) How much salt is in the tank after t minutes? (b) What is the ”terminal ” salt amount in the tank when t → ∞? 1 ...
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