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Unformatted text preview: Diﬀerential Equations test one Tuesday, April 25, 2006 x 1. Given the function y = 4+
2 sin s2 ds (b) Calculate
dy . dx (a) Calculate y (2). 2. Is the function y = x+ 1 dy 2 a solution of +y = 3+x2 ? x dx 3. Determine whether each of the following equations is linear or nonlinear. dy = t − y2 dt dy = et y −cos t2 dt dy = 4t − y dt (a) (b) t (c) y 4. Which of the following equations is exact? (a) 3x2 − 4y dx+(2y − 4x) dy = 0 dy + x2 y = x dx (b) 5. Given the initial value problem t dy 3 4 + y= and y (3) = 5 dt t − 2 t+1 determine the largest tinterval over which a unique solution is guaranteed to exist. 6. Given the initial value problem dy √ = y 3 + x 3 y and y (3) = 0 dx can one be certain that a unique solution exists? page two 7. Use Euler’s method (i.e. the tangentline method) with stepsize h = 1/2 to ﬁll in the following table for the initial value problem dy = 2x − y 2 and y (1) = 2 dx y dy dx approximate ∆y x 1 1.5 2 8. Solve each of the following diﬀerential equations or initial value problems: dy = 2xey dx dy dt (a) (b) + 2y = 5e3t y (0) = 3 (c) 3x2 − 4y dx+(2y − 4x) dy = 0 9. The rate of population growth of a small country is proportional to the country’s population. At time t = 0 years the country’s population is equal to 5 million. (a) From time t = 0 years to time t = 20 years the population grows at an instantaneous rate equal to 4% of the population. Set up the diﬀerential equation for the country’s population P (t) and solve it. (b) Starting at time t = 20 years the population grows at an instantaneous rate equal to only 2% of the population. Set up the diﬀerential equation for the country’s population during this interval and solve it. ...
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 Spring '09
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