lec_10w - August 18, 2010 10-1 10. Particular Solutions of...

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Unformatted text preview: August 18, 2010 10-1 10. Particular Solutions of Non-homogeneous second order equationsVariation of Parame- ters As above, we consider the second order differential equa- tion L ( y ) = y 00 + p ( t ) y + q ( t ) y = g ( t ) (1) where p, q, g are continuous functions in an interval I . In the method called variation of parameters, we start with two linearly independent solutions y 1 , y 2 of L ( y ) = 0, and we look for a particular solution y ( t ) of L ( y ) = g of the form y ( t ) = y 1 ( t ) v 1 ( t ) + y 2 ( t ) v 2 ( t ) (2) where v 1 and v 2 are not constant functions. It turns out that we can consider the system of linear equations y 1 v 1 + y 2 v 2 = 0 (3) y 1 v 1 + y 2 v 2 = g ( t ) (4) We solve these equations for v 1 and v 2 and integrate to get v 1 and v 2 . August 18, 2010 10-2 Let us verify that if the equations (3)and (4) are sat- isfied, then (2) does indeed give us a particular solution. Let us begin by observing that differentiating the first equation above gives ( y 1 v 1 + y 2 v 2 ) = y 1 v 1 + y 2 v 2 + y 1 v 00 1 + y 2 v 00 2 = 0 (5) Hence, using (3), (4), and (5), we get y 00 + py + qy = ( y 1 v 1 + y 2 v 2 ) 00 + p ( y 1 v 1 + y 2 v 2 ) + q ( y 1 v 1 + y 2...
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This note was uploaded on 02/03/2012 for the course MTH 235 taught by Professor Staff during the Fall '11 term at Michigan State University.

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lec_10w - August 18, 2010 10-1 10. Particular Solutions of...

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