Derive gen sol of non-homogeneous DEs through linear algebra

[Kevin Qi]

Hello,
I noticed that the solution of a homogeneous linear second order DE can be interpreted as the kernel of a linear transformation.
It can also be easily shown that the general solution, Ygeneral, of a nonhomogenous DE is given by:
Ygeneral = Yhomogeneous + Yparticular

My question: Is it possible to arrive at the above result by using arguments involving the relationship between the kernel, column space, row space, etc of the linear differential operator of a DE?

My current attempt: The solution space of a non-homogeneous DE is isomorphic to the solution space (kernel) of the corresponding homogeneous DE, and is "displaced" from the kernel by the vector, Yparticular. Hence Ygeneral could be constructing by adding any Yparticular back to the Yparticular.

My solution is probably pretty flawed, and I have no idea how I can justify it. Could someone please enlighten me (or tell me that my question is dumb)? :D

Thanks in advance!

 

[HallsofIvy]

The set of all solutions to an nth order linear homogeneous equation is an n dimensional vector space. That can be visualized as an n dimensional "hyper-plane" containing the origin. If v is a specific solution then adding v to every member of that hyper-plane gives a hyper-plane that does not contain the origin.

 

[Kevin Qi]

Hi HallsofIvy,
Thanks for taking the time to help me!
Just to make sure that I got this right: Can you say that the specific solution vector is the "Pseudo-Origin" of the non-homogeneous equation' solution space, in the same way the 0 vector is the origin of the homogeneous equation's solution space?