Particulate solution for $y''(x)-3y'(x)=2x+1+e^{3x}$ I am in the process of solving this differential equation: $$y''(x)-3y'(x)=2x+1+e^{3x}$$
I have the homogeneous solution. However, I am not getting anywhere with the particular solution.
One of the homogeneous solutions is $e^{3x}$, which is why we would multiply an $x$ to it in the particulate approach. However, the other homogeneous solution is $1$ and I don't know exactly how it works.
Could someone help me with this?

Hint: Three ways:

*

*Let $w=y'$ so you reduce it to a first order differential equation $w'-3w=x+1+e^{3x}$


*Use the method called "Undetermined Coefficients". Here is a link:
https://tutorial.math.lamar.edu/classes/de/undeterminedcoefficients.aspx
This is a standard method to find particular solutions under some conditions. You can see the table in this link, and it helps organize those cases.


*Use the method called "Variation of Parameters". Here is a link:
https://tutorial.math.lamar.edu/classes/de/VariationofParameters.aspx
This is also a standard method to find particular solutions, and it is much more powerful than method #2.
