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Consider the following differential equation:

\(\displaystyle {y}''+{y}'= x^{2}\)

I have found the homogeneous solution to be:

\(\displaystyle y_{H}=c_{1} + c_{2}e^{-x}\)

But when finding the particular solution, using reduction of order, I end up getting:

\(\displaystyle y_{P}=\frac{x^{3}}{3} + \frac{cx^{2}}{2} + dx + e\)

By substituting the results for \(\displaystyle {y}''\) and \(\displaystyle {y}'\) back into the original equation, I am able to obtain \(\displaystyle c = -2\) and \(\displaystyle d = 2\). But what do I do about \(\displaystyle e\)?