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Math 314 W.T. Kiley Exam 1 February 13-20, 2001 This part of the exam is take home open book. You may not discuss it with anyone. It is due at 10:30 a.m. on February 20. Clearly show your work and your answers. 2 1. Find a basis for the solutions of x y" -3xy' + 4(x + 1)y = 0 around x = 0. Find a general term for one member or your basis and the first 3 terms of the other. 2. Do 2 of the special cases in 4.4 # 16c. 3. 4.4 #20. 4. 4.5 # 8 5. 4.6 # 12, the part about approximating the zeros. 6. Find J (2) by using the recurrence relations in the text and A1 on page A85. 5 MAth 314 February 15, 2001 This part of the exam is closed book. You may use one page of notes. 2 2 2 a) x y" + xy' - y = 0; b) y" + xy' - x y = 0; c) x y" + (3x - 1)y' + y = 0. 1. One of these has an analytic solution satisfying y(0) = 1, y'(0) = 2. Find the terms of the series through x to the 4'th. 2. One of these can be solved near 0 by the Frobenius method. Use the method to find a basis for the solution space. 3. Use the method for the remaining equation. Find the radius of convergence. 4. Differentiate the power series of J . 0 5. Show that between 2 zeros of J 0 Compare the result with J . 1 there is a zero for J . 1 2 2 6. Use the change of variable z = 2x to solve x y" + xy' + (4x - 9)y = 0 in terms of Bessel functions. x 7. Assume e and sin x are a basis for the solution space of an equation. 2 2 a) Can 2 + 3x + 4x be the first terms of a solution? b) Can 2 + 3x + x ?