Solve using the annihilator approach of undetermined coefficients.

y'' + 3y' -10y = (x+1)e^(x)sinx

Question

Solve using the annihilator approach of undetermined coefficients. 

y'' + 3y' -10y = (x+1)e^(x)sinx

anonymous · Answer

Every attempt takes a lot of time and I just need to know for sure what "guess" I should even be using for a particular solution before I embark on another waste of time doing mass differentiation :/

anonymous · Answer

$$y_{c} = c_{1}e^{-5x}+ c_{2}e^{2x}$$ That's the easy part. I think the operator for the f(x) portion would be

$$(D^{2}-2D + 2)^{2}$$ meaning the missing roots are $$1 \pm i, 1 \pm i$$ if these were a part of yc, this would turn into:
$$e^{x}(cosx + sinx) + xe^{x}(cosx + sinx)$$But then I'm supposed to base my "guess" off of that, right? Well, a guess for e^(x)cosx by itself would be Ae^(x)cosx + Be^(x)sinx. Would I really have to do that for all 4 terms? e^(x)cosx + e^(x)sinx + xe^(x)cosx + xe^(x)sinx?? Wouldn't that make like 10 constants in my guess? T_T