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MATH 266 Section A1 Spring 2013 PRACTICE FINAL EXAM 10 points each 1. Find the general solution of y 0 = 2y 2 + 3. 2. Solve the initial value problem y 0 + y x+1 = x2 y(1) = 2. 3. Find the general solution of y 000 + 6y 00 + 9y 0 = t − 3et . 4. Solve the initial value problem y 00 − 8y 0 − 20y = 0 5. Find the general solution of y 0 = y(0) = 5 y 0 (0) = 14. y 2 −xy x2 6. 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. 7. Find the solution of 2 2 x = x 1 3 0 8. Let 0 x(0) = 2 1 −1 −1 0 A = 1 1 3 0 1 Find a fundamental set for x0 = Ax. 9. Find all equilibrium solutions of the system x0 = xy y 0 = −x + 2y + 4 and classify each by type and stability. 10. Show that {1 + t, et } is a fundamental set for ty 00 − (t + 1)y 0 + y = 0 Then use the variation of parameters method to find a particular solution of ty 00 − (t + 1)y 0 + y = t2 et