I have been given the question $y'' - 9y' = 9e^{9x}$ to solve. Per my knowledge, this is a second order non-homogeneous differential equation. By using the method of undetermined coefficients, I am supposed to find the solution of the homogeneous $+$ the particular solution of the non homogeneous. The homogeneous led me to $y = c_1 + c_2e^{9x}$.
How do I proceed from this point?
For the particular solution, try : $$y_p=Axe^{9x}$$
Another way $$y'' - 9y' = 9e^{9x}$$ $$y''e^{-9x} - 9y' e^{-9x}= 9$$ $$(y'e^{-9x})'= 9$$ Integrate $$(y'e^{-9x})= 9x+K_1$$ $$y'=e^{9x}(9x+K_1)$$ Integrate again $$y=9\int e^{9x}xdx+K_1e^{9x}+K_2$$ $$\boxed{y(x)=e^{9x}x+K_1e^{9x}+K_2}$$