variation_of_parameters

# variation_of_parameters - Math 107 Variation of Parameters...

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Math 107 Variation of Parameters October 5, 2005 The ordinary differential equation L ( y ) q n ( x ) y ( n ) ( x ) + q n - 1 ( x ) y ( n - 1) ( x ) + . . . + q 1 ( x ) y ( x ) + q 0 ( x ) y ( x ) = g ( x ) has solutions of the form y ( x ) = y p ( x ) + c 1 y 1 ( x ) + . . . c n y n ( x ) where y 1 , . . . y n are linearly independent solutions of the homogeneous equation L ( y i ) q n ( x ) y ( n ) i ( x ) + q n - 1 ( x ) y ( n - 1) i ( x ) + . . . + q 1 ( x ) y i ( x ) + q 0 ( x ) y i ( x ) = 0 , 1 i n Here is a way to use y 1 , . . . y n and g to find y p . Define Y ( x ) = y 1 ( x ) y 2 ( x ) . . . y n ( x ) y 1 ( x ) y 2 ( x ) . . . y n ( x ) . . . . . . . . . y ( n - 1) 1 ( x ) y ( n - 1) 2 ( x ) . . . y ( n - 1) n ( x ) and compute the n vector u ( x ) = x Y ( s ) - 1 e n g ( s ) q n ( s ) ds . Here e n is the n -th axis vector, with zeros in the first n entries and one in the last. Then y p ( x ) = e 1 Y ( x ) u ( x ) is a particular solution of the inhomogeneous ordinary differential equation. In this expression, e 1 is the first axis vector, with one in the first entry and zeros in the last n - 1 entries. To prove this fact, we note that the definition of y p ( x ) can be rewritten y p ( x ) = n j =1 y j ( x ) u j ( x ) .

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