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$\frac{\mathrm{d^2}y }{\mathrm{d} x^2} - 2\frac{\mathrm{d} y}{\mathrm{d} x} + 2y = 4e^xsinx$

Then my complementary function:

$y = e^x(c_{1}cosx + c_{2}sinx])$

I know that $e^xsinx$ is in the complementary function but I don't know how to address that in this example

I've re-written $4e^xsinx = 2(e^{(i+1)x} - e^{(1-i)x})$

Then I have $y_{PI_{1}} = Ae^{(i+1)x}$ and $y_{PI_{2}} = Be^{(1-i)x}$ with $y_{PI} = y_{PI_{1}} + y_{PI_{2}}$

But then I get stuck, and I'm not sure if this is the correct way. The solution provided says to take $y_{PI} = Axe^{(1+i)x}$ aiming to take the imaginary part later. I don't understand what it means/how that would work, if someone could provide a slightly more detailed explanation that would help.

Sorry if my question is confusing, new to the topic.

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    **Hint:** Choose $y_p(x) = x e^x( a \cos x + b \sin x)$. We add the $x$ because we found $e^x \sin x$ in the complementary solution. – 2017-02-19
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    Sorry for the long response time, had dinner. I used what you said and arrived at the correct answer so thanks for the help. That being said, the differentiation required was long winded and took me some time. Just wondering if this is the most efficient method? – 2017-02-19
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    You could try Laplace Transforms. – 2017-02-19

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