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Unformatted text preview: Diﬀerential Equations test one Monday, April 25, 2005
please show your work in order to get full credit 1. Which of the following diﬀerential equations is linear? (a) d2 x = − sin x dt2 (b) x2 y −(tan x)y = ex (c) dP = P − P2 dt 2. For which values of t is the solution to the following diﬀerential equation guaranteed to exist? (t − 2)(t + 1)y + ty = √ 25 − t and y (4) = −2 3. Given the initial value problem dy dx = 4x − 2 y y (0) = 1 (a) Is a unique solution to this initial value problem guaranteed to exist near x = 0? (b) Use Euler’s method with step size h = 0.1 to ﬁll in the following table: xy 01 0.1
dy dx approximate ∆y (c) Write down an equivalent integral equation. (d) In the method of successive (Picard) approximations to the solution, a ﬁrst (constant) solution of y0 (t) = 1 is chosen. What is the next approximate solution? 4. Solve each of the following diﬀerential equations or initial value problems. x dy − 2y = 4x2 dx (12x2 − 16xy )dx + (4y 3 − 8x2 )dy = 0 (a) dv 4x =− dx v (b) (c) y (1) = 5 y (1) = 1 5. Given the equation dy = −y (y − 2)(y − 99), ﬁnd each equilibrium (constant) solution and dt discuss the stability of each equilibrium solution. continue to page two page two 6. A large 5000 liter tank contains 3000 liters of pure water at time t = 0. Beginning at time t = 0, salt water is pumped into the large tank at a rate of 20 liters per second. Each liter of this salt water contains 30 grams of salt. Let S (t) represent the amount of salt in the large tank at time t. (a) For the ﬁrst 100 seconds, the large tank is ﬁlling up. Write a diﬀerential equation for S (t) when 0 ≤ t < 100, and solve it. (b) After the ﬁrst 100 seconds, the tank overﬂows. Write a diﬀerential equation for S (t) for t ≥ 100, and solve it. ...
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 Spring '09
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