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week07b - Week 7a 4.3 Undetermined Coef[Tables TABLE of...

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Week 7a: 4.3 Undetermined Coef [Tables] TABLE of trial y p : Usual trial y p F ( x ) Usual Modified ce ax y p = A 0 e ax ce ax cos( bx ) or y p = e ax ( A 0 cos( bx )+ ce ax sin( bx ) B 0 sin( bx )) cx k y p = A 0 + A 1 x + · · · + A k x k When: The root associated with F ( x ) is NOT a root of the characteristic polynomial P ( r ) giving y H in the general solution y = y H + y p . In the 3 cases, P ( a ) = 0; P ( a + bi ) = 0; and P (0) = 0 . Applications often feature the “pure imaginary” complex case, where a = 0 , for which we get the usual solution when P ( ib ) = 0 . ———————- TABLE of trial y p : Simple modified solution F ( x ) Usual Modified ce ax y p = A 0 e ax y p = A 0 xe ax ce ax cos( bx ) or y p = e ax ( A 0 cos( bx )+ y p = xe ax ( . . . ) ce ax sin( bx ) B 0 sin( bx )) cx k y p = A 0 + A 1 x + · · · + A k x k y p = x ( . . . ) When: The root associated with F ( x ) is a simple root of the characteristic polynomial P ( r ) . If the DE was y + a 1 y + a 2 y = F, so P ( r ) = r 2 + a 1 r + a 2 , the cases are (1) P ( r ) = 0 has distinct real roots, one of which is r 1 = a ; (2) P ( r ) = 0 has complex roots a ± bi, the SAME complex roots
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