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UndeterminedCoefs

UndeterminedCoefs - More on the method of undetermined...

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More on the method of undetermined coefficients February 26, 2008 More on the method of undetermined coefficients
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Example y + y = cos t Using operator notations this is ( D 2 + 1) y = cos t . We have already seen a version of this problem in our exploration of the driven spring oscillator. Let’s use the annihilator method to find the solution. To annihilate the cosine, which is associated to the roots ± i , we need to apply D 2 + 1 = ( D + i )( D - i ) to both sides of the equation, and we get ( D 2 + 1) 2 y = ( D 2 + 1) cos t = 0 . This is the case of a complex conjugate pair of roots ( ± i ) repeated twice, and the solution is y = C 1 cos t + C 2 t cos t + C 3 sin t + C 4 t sin t . Note that the homogeneous solution corresponding to the original equation is y h = C 1 cos t + C 3 sin t , so we can take y p = C 2 t cos t + C 4 t sin t . More on the method of undetermined coefficients
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Solving y + y = cos t y p = C 2 t cos t + C 4 t sin t . We plug this into the original equation and we get y p + y p = ( C 2 ( - 2 sin t - t cos t ) + C 4 (2 cos t - t sin t )) + C 2 t cos t + C 4 t sin t = - 2 C 2 sin t + 2 C 4 cos t = cos t leading to - 2 C 2 sin t + 2 C 4 cos t = cos t , and C 2 = 0, C 4 = 1 2 . Putting this together we get y = y h + y p = C 1 cos t + C 3 sin t + 1 2 t sin t , where the values of C 1 and C 3 can be determined by initial conditions. More on the method of undetermined coefficients
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Better than Rules A more general rule has to take into account the possibility that the root introduced when you annihilate the forcing function may have already been present as a root of the original operator. On page 184 of our text, there is such a rule. I have trouble remembering it. However, I can remember the rule for solving higher order homogeneous equations,and if I can recognize what operators I need to apply to annihilate the forcing function, it is easy to identify the right form with undetermined coefficients —it will be the terms in the general solution of the high order homogeneous equation that are not in y h . More on the method of undetermined coefficients
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Finding annihilators Identifying the right operators to use to annihilate is the reverse of the process that we used to solve the homogeneous equations. If we have an e rt we use D - r , if we have p ( t ) e rt where p ( t ) is a polynomial of degree k , then we use ( D - r ) k +1 .
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