General Higher Order Linear Non-Homogeneous Diff Eq's

[SrVishi]

Hello, I am learning about the general solution to higher order linear non-homogeneous differential equations. I know that the general solution of such an equation is of the form $y=y_h+y_p$ where $y_h$ is the solution to the respective homogeneous equation and $y_p$ is a particular solution of the equation. I also realize that in the homogeneous solution, you want to include as many terms as possible, even giving a parameter of constants to each of these terms to include as many different solutions as possible. My question is, why do we only need one particular solution? If we are trying to make our general solution as "general" as possible, why do we not include a family of solutions that satisfy the non-homogeneous differential equation?

 

[lurflurf]

The homogeneous solutions are differences of the particular solutions.
yh=yp2-yp1
if we have
one yp and all yh
we have all solutions
say we are worried we do not have yp2
yp2=yp1+(yp2-yp1)
but we have yp1 and (yp2-yp1) is homogeneous so we have it
so we had yp2 all along