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Unformatted text preview: then, we have the final solution: 25 Step 1 Write the given differential equation y" + xy' + 2y = 0 (1) where y(0) = 1, y'(0) = 0 Step 2 Let  (2) Step 3 differentiating y with respect to x y' = Step 4 do the same to y': y" = Step 5 Substituting y, y', y" into (1), we have:  (3) Step 6 note: , we can conbine the second and third term in (3):  (4) Step 7 Replace n by n+2 in the first term of (4), and starting sum at 0, (4) becomes:  (5) Step 8 combine all term in (5) together: the coefficients of both side of the equation must be equal, so : [(n+2)(n+1)a n+2 + (n+2) a n ] = 0 , n = 0,1,2,. .. or , n=0,1,2,. ..  (6) Step 9 since y(0) = 1 = 1, we have: a = 1; since y'(0) = 0: we have: a 1 = 0; Step 10 use a = 1, and a 1 = 0, we have the following: and so on, note: when n is odd, a n is zero Step 11 Substitue back into (2), we have: then, we have:...
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This note was uploaded on 09/25/2011 for the course MATHS 12 taught by Professor Ramarao during the Fall '10 term at Punjab Engineering College.
 Fall '10
 RamaRao
 Math, Differential Equations, Equations

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