ELEMENTRY DIFFERENTIAL EQUATIONS 5.2

# ELEMENTRY DIFFERENTIAL EQUATIONS 5.2 - then, we have the...

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ELEMENTRY DIFFERENTIAL EQUATIONS 9 th Edition, Boyce Chapter 5.2 1- 3- 5- 7- 9- 11- 13-

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15- 17- 19- 21- 23- Step 1 Write the given differential equation y" - xy' - y = 0 --(1) where y(0) = 1, y'(0) = 0 Step 2 Let --- (2) Step 3 differentiating y with respect to x y' = -- (3) Step 4 do the same to y':
y" = -- (4) Step 5 Substituting y, y', y" into (1), we have: -- (5) note: , we can conbine the second and third term in (5): -- (6) Step 6 Replace n by n+2 in the first term of (6), and starting sum at 0, (6) becomes: -- (7) Step 7 combine all term in (7) together: Step 8 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,. .. -- (8)

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Step 9 since y(0) = 1 = 1, we have: a 0 = 1; Step 10 since y'(0) = 0: we have: a 1 = 0; Step 11 use a 0 = 1, and a 1 = 0, we have the following: and so on, note: when n is odd, a n is zero Step 12 Substitue back into (2), we have:

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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.

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ELEMENTRY DIFFERENTIAL EQUATIONS 5.2 - then, we have the...

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