SampleTest3v2

SampleTest3v2 - MATH3705 Solution to Test 3 Summer 2006 Solution to Test 3 MAT 3705 Summer 2006 1(5 marks Consider initial-value problem y xy 2y =

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MATH3705 Solution to Test 3 Summer 2006 1 Solution to Test 3 MAT 3705, Summer 2006 1. (5 marks) Consider initial-value problem y" + xy' + 2 y = 0, y (0) = 1, y' (0) = 0. Find the first four non-zero terms of a series solution of this problem about x = 0. Solution . Since x = 0 is an ordinary point of this equation, we may let y = a 0 + a 1 x + a 2 x 2 + … . y (0) = a 0 = 1, y' (0) = a 1 = 0. y' = a 1 + 2 a 2 x + 3 a 3 x 2 + … y" = 2 a 2 + (3 × 2) a 3 x + (4 × 3) a 4 x 2 + (5 × 4) a 5 x 3 + … xy' = a 1 x + 2 a 2 x 2 + 3 a 3 x 3 + … 2 y = 2 a 0 + 2 a 1 x + 2 a 2 x 2 + 2 a 3 x 3 + … Hence, a 2 = - a 0, a 3 = - 3 a 1 / (3 × 2), a 4 = - 4 a 2 / (4 × 3), a 5 = - 5 a 3 / (5 × 4). In general, a n = - na n - 2 / [ n ( n - 1)] = - a n - 2 / ( n - 1). Since a 1 = 0, a 3 = a 5 = … = a 2 k +1 = … = 0. Since a 0 = 1, a 2 = - a 0 = - 1, a 4 = - a 2 / 3 = 1 / 3, a 6 = - a 4 / 5 = - 1 / 15. The first four non-zero terms are y = 1 - x 2 + (1 / 3) x 4 - (1 / 15) x 6 + … .
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This note was uploaded on 07/16/2009 for the course MATH 3705 taught by Professor Jaberabdualrahman during the Winter '08 term at Carleton CA.

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SampleTest3v2 - MATH3705 Solution to Test 3 Summer 2006 Solution to Test 3 MAT 3705 Summer 2006 1(5 marks Consider initial-value problem y xy 2y =

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