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m2k_opm_cnvapl

# m2k_opm_cnvapl - Applications of Convolution The Variation...

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Applications of Convolution The Variation of Parameters Formula In the theory of linear inhomogeneous differential equations it is shown that the particular solution y ( x ) of the differential equation a 0 d n y dx n + a 1 d n - 1 y dx n - 1 + · · · + a n - 1 dy dx + a n y = f ( x ) with zero initial conditions, i.e., y (0) = 0 , y 0 (0) = 0 , · · · , y ( n - 1) (0) = 0 , is given by the so-called Variation of Parameters formula: y ( x ) = Z x 0 W ( x, ξ ) f ( ξ ) dξ, wherein the kernel function, W ( x, ξ ), is given in terms of a cer- tain Wronskian determinant. This determinant is rather labori- ous to compute for even modest values of n , such as 4 or 5. Using the Laplace transform we can obtain a very simple formula for W ( x, ξ ) which is computable with very modest effort. Application of the Laplace transform to the differential equa- tion above gives p ( s ) ( L y ) ( s ) = ( L f ) ( s ); p ( s ) = a 0 s n + a 1 s n - 1 + · · · + a n - 1 s + a n . Then ( L y ) ( s ) = 1 p ( s ) ( L f ) ( s ) . 1

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Suppose p ( s ) has distinct roots λ k , k = 1 , 2 , · · · , n ; real or complex. Then we can write 1 p ( s ) = c 1 s - λ 1 + c 2 s - λ 2 + · · · + c n s - λ n = n X k =1 c k s - λ k , where the c k are given by the residue formula c k = 1 p 0 ( λ k ) . Since the Laplace transform carries convolution products into ordinary products of functions (of s ), the inverse Laplace trans-
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m2k_opm_cnvapl - Applications of Convolution The Variation...

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