VariationParameters.pdf - Method of Variation of parameters...

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Unformatted text preview: Method of Variation of parameters S. Bonnot February 15, 2011 S. Bonnot Method of Variation of parameters Setting of the method Goal : Find a particular solution for y 00 + p ( t ) . y + q ( t ) . y = g ( t ) where the right-hand side can be an arbitrary continuous function. Example : y 00- 3 y + 4 y = ( tan t ) . ( t 2 + cos ( t )) observe that the right-hand side is not in our list of simple functions... Main hypothesis : assume that the general solution of y 00 + p ( t ) . y + q ( t ) . y = 0 is y ( t ) = c 1 y 1 ( t ) + c 2 y 2 ( t ) Main Idea: Try to find a particular solution y p of y 00 + p ( t ) . y + q ( t ) . y = g ( t ) in the following form y p = c 1 ( t ) y 1 ( t ) + c 2 ( t ) y 2 ( t ) The parameters are varying S. Bonnot Method of Variation of parameters Description of the method Start with y p = c 1 ( t ) y 1 ( t ) + c 2 ( t ) y 2 ( t ) Line L1 Then y p = c 1 y 1 + c 2 y 2 + c 1 ( t ) . y 1 + c 2 . y 2 We set c 1 ( t ) . y 1 + c 2 . y 2 = and call this Equ.1 observe that now y p = c 1 y 1 + c 2 y 2 Line L2 Derive again: y 00 p = c 1 y 00 1 + c 2 y 00 2 + c 1 y 1 + c 2 y 2 Line L3 Now, do (L3)+p(t).(L2)+q(t).(L1)Now, do (L3)+p(t)....
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VariationParameters.pdf - Method of Variation of parameters...

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