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MATH 267 Section J Fall 2012 PRACTICE FINAL EXAM 1. (20 points) Solve the initial value problem: (a) y 0 = 1 xy y(1) = 3. (b) y 00 − 8y 0 − 20y = 0 y(0) = 5 y 0 (0) = 14. 2. (20 points) Find the general solution: (a) y 0 + y = x+1 00 x2 . (b) y 000 + 6y + 9y 0 = x − 3ex . 3. (10 points) Solve by the Laplace transform method y 00 + 6y 0 − 7y = u4 (t) y(0) = 1 y 0 (0) = 2 4. (10 points) FindPthe recurrence relation and the first 5 nonzero terms in the power series n solution y(x) = ∞ n=0 an x of y 00 + y 0 − xy = 0 y(0) = 2 y 0 (0) = 1 5. (10 points) Census Bureau estimates of the Unites States population are 281.4 million for April 1, 2000 and 308.7 million for April 1, 2010. Assuming that population grows at a rate proportional to the population, estimate the United States population on April 1, 2020 to the nearest million. 6. (10 points) Find the general solution of the system −3 2 0 x = x 2 −3 and sketch the phase portrait. 7. (10 points) Find all equilibrium solutions of the system x0 = xy y 0 = −x + 2y + 4 and classify each by type and stability. 8. (10 points) Show that {1 + x, ex } is a fundamental set for xy 00 − (x + 1)y 0 + y = 0 Then use the variation of parameters method to find a specific solution of xy 00 − (x + 1)y 0 + y = x2 ex (Remember to write the equation so that the coefficient of y 00 is 1.)