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Test 3 Solutions

Test 3 Solutions - MAT 274 TEST 3 Instructor Dongrin Kim...

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MAT 274 TEST 3 Instructor : Dongrin Kim Name : Score : 1. Find the radius of convergence of the power series X n =0 ( - 1) n n 2 ( x + 2) n 3 n . Ans : 3 2. Determine the Taylor series about the point x 0 . Also determine the radius of convergence of the series. (a) e - 2 x , x 0 = 0 Ans : X n =0 ( - 1) n 2 n n ! x n , ρ = (b) 2 x ln(1 - 3 x ) , x 0 = 0 Ans : - X n =0 6 · 3 n n + 1 x n +2 , ρ = 1 3 1
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3. Rewrite the expression as a sum whose generic term involves x n . (a) X n =0 a n x n +2 Ans : X n =2 a n - 2 x n (b) X n =1 na n x n - 1 + x X n =0 a n x n Ans : X n =0 ( n + 1) a n +1 x n + X n =1 a n - 1 x n 4. Determine a lower bound for the radius of convergence of the series solution about x = 3 for the following differential equations. (a) ( x 2 + 2 x + 2) y 00 + xy 0 + 4 y = 0. Ans : 17 (b) y 00 + (sin x + 1) y 0 + (cos x ) y = 0 Ans : (c) ( x 2 - 1) y 00 + 2 xy 0 + 3 y = 0 Ans : 2 2
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5. Find the series solution about x 0 = 0 for y 00 + xy 0 + 2 y = 0. (a) Find the recurrence relation. Ans : a n +2 = - a n n + 1 , n 0 (b) Find the first three terms in each of two linearly independent solutions. Ans : y 1 = 1 - x 2 + x 4 3 + · · · y 2 = x - x 3 2 + x 5 10 + · · · 6. Find all singular points and determine whether each one is regular or irregular
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