For the DE $ y”+9y’+9y=x^3e^{4x}$ , I’m confused what my trial solution should be for the particular integral. I know that if the RHS is $ P_n(x)e^{\lambda x}$ and $ m_1, m_2$ solution of the characteristic equation, then the trial solution is normally $ x^kQ_n(x)e^{\lambda x}$ , where:

$ k=0$ if $ m_1 \neq m_2 \neq \lambda$

$ k=1$ if $ m_1 \neq m_2 = \lambda$

and $ k=2$ if $ m_1=m_2=\lambda$ .

But my situation is the case $ m_1=m_2 \neq \lambda$ .

Intuitively I feel like I should try $ Q_{n+1}(x)e^{\lambda x}$ , but I have no idea if this is correct – or, if it is, why.