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Unformatted text preview: Series sol. near ordinary points w/ given initial conds. Lets revisit the 2 problems that we have solved in class but lets solve them with some initial conditions. We will solve them two using two different methods. 1. Using the results we obtained in ordinary_ps.pdf . 2. Using a Taylor series expansion of the solution about x . Before doing so, recall that the Talor series expansion of a function about point x is given by y ( x ) = y ( x ) + y ( x )( x x ) + y 00 ( x ) 2! ( x x ) 2 + y 000 ( x ) 3! ( x x ) 3 + y (4) ( x ) 4! ( x x ) 4 + . (1) For a given second order initial value problem (IVP), we already know y ( x ) and y ( x ) . What we need to find are y 00 ( x ) ,y 000 ( x ) , . We will find these using the given differential equation, as illustrated in the problems below. 1 c HF, 2011 Series sol. near ordinary points w/ given initial conds. Problem 1 (1 + x ) y 00 + y = 0 , x = 0 . y (0) = 1 ,y (0) = 1 . (2) Solution Method A : Recall that we previous found the general solution to the problem to be (see ordinary_ps.pdf ) y = a 1 1 2 x 2 + 1 6 x 3 1 24 x 4 + + a 1 x 1 6 x 3 + 1 12 x 4 + , (3) where a and a 1 are two arbitrary constants. Using the first initial conditions, we find y (0) = 1 = a 1 1 2 2 + 1 6 3 1 24 4 + + a 1 1 6 3 + 1 12 4 + a = 1 ....
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 Spring '10
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