$$4y''+16y'+17y=e^{-2t}\sin(\frac{t}{2})+6t\cos(\frac{t}{2})$$
Shouldn't it the guess be:
$$y_{p}=e^{-2t}\left(A\sin(\frac{t}{2})+B\cos(\frac{t}{2})\right)+(Ct+D)\left(\cos(\frac{t}{2})+\sin(\frac{t}{2})\right)$$?
particular solution of $4y''+16y'+17y=e^{-2t}\sin\frac{t}{2}+6t\cos\frac{t}{2}$
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ordinary-differential-equations
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0As long as the coefficients are solvable, the guess should be correct. – 2017-02-06
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0See https://www.wolframalpha.com/input/?i=4y%27%27%2B16y%27%2B17y%3De%5E%7B-2t%7D%5Csin(%5Cfrac%7Bt%7D%7B2%7D)%2B6t%5Ccos(%5Cfrac%7Bt%7D%7B2%7D). – 2017-02-06
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0@hkmather802 http://tutorial.math.lamar.edu/Classes/DE/UndeterminedCoefficients.aspx#Second_UnderCoeff_Ex10c gave a different answer – 2017-02-06
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0@gbox you can use Laplace transform to solve this! – 2017-02-06
1 Answers
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Note that the solutions of the characteristic equation are $-2\pm{i\over2}$. It follows that $e^{-2t}\sin{t\over2}$ is a solution of the corresponding homogeneous equation. You should take care of this in your "Ansatz" for $y_p$.