How to use undetermined coefficients for right hand side of this differential equation: y''+y'+8y = t*cos3t + (10t^2 +21t + 9) * sin3t

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How to use undetermined coefficients for right hand side of this differential equation: y''+y'+8y = t*cos3t + (10t^2 +21t + 9) * sin3t

Mathematics
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This is a large problem to write out online, but you can tackle it by finding the particular solution as such. You can find particular solutions to each of\[y''+y'+8y=t \cos 3t\]and\[y''+y'+8y=(10t^2+21t+9) \sin 3t\]and add them to find the particular solution to the whole thing. To find the particular solution for the first equation set,\[y_1=(At+B)(C \cos 3 t + D \sin 3t)\]and for the second,\[y_2=(At^2+Bt+C)(D \cos 3t + E \sin 3t)\]Substitute each into your d.e. and find the coefficients. Your particular solution will then be\[y_p=y_1+y_1\]If you need any further help, let me know.
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Thank you so much,lokisan. What I didn't get was, if I should write underdetermined coefficients for both polynomial and trigonometric function or just for trigonometric function -which are in multiplication- [ex. in my textbook (e^t * sint) is written in underdetermined coefficients as [e^t * (A*cost + B*sint)] Now I get i should write for both and look for if there are common coefficients (like if there are two A*D coefficient I could write A to simplify that,hope I got it right.

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oktal, you wasting time here and not studying?!
oktal,yep,for sure!

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