Undetermined Coefficients and Variations of Parameters

Undetermined Coefficients and Variations of Parameters -...

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Unformatted text preview: Undetermined Coefficients for y + p(x)y + q(x)y = g(x) g(x) (1) (2) (3) (4) (5) pn (x) = an xn + . . . + a1 x + a0 aex a cos x + b sin x pn (x)ex pn (x) cos x + qm (x) sin x, where qm (x) = bm xm + . . . + b1 x + b0 aex cos x + bex sin x pn (x) ex cos x + qm (x) ex sin x yp (x) xs Pn (x) = xs (An xn + . . . + A1 x + A0 ) xs Aex xs (A cos x + B sin x) xs Pn (x)ex xs {PN (x) cos x + QN (x) sin x}, where QN (x) = Bn xN + . . . + B1 x + B0 and N = max(n, m) xs (Aex cos x + Bex sin x) xs ex {PN (x) cos x + QN (x) sin x}, where N = max(n, m) (6) (7) The nonnegative integer s is chosen to be the least integer such that no term in yp (x) is a solution of the corresponding homogeneous equation y + p(x)y + q(x)y = 0. Variation of Parameters If y1 are y2 are two linearly independent solutions of y +p(x)y +q(x)y = 0, then a particular solution of y + p(x)y + q(x)y = g(x) is y = v1 y1 + v2 y2 , where v1 (x) = -g(x)y2 (x) dx, W [y1 , y2 ](x) v2 (x) = g(x)y1 (x) dx, W [y1 , y2 ](x) and W [y1 , y2 ](x) = y1 (x)y2 (x) - y1 (x)y2 (x). ...
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