Hi all,

I'm currently working on a question that I can't fully work out. Can someone look over my working and see where I may have possibly gone wrong and maybe point me in the right direction?

The question is,

Given a LDE $\displaystyle F=b_{0}y''+b_{1}y'+b_{2}y=0$ and the relationship $\displaystyle b_{0}''-b_{1}'+b_{2}=0$, show that the factorization of the LDE is $\displaystyle F=D[b_{0}D+(b_{1}-b_{0}')]y$.

From the relationship, I can use $\displaystyle b_{2}=b_{1}'-b_{0}''$ and I can factorize that into $\displaystyle b_{2}=D[b_{1}-b_{0}']$.

Factorizing: F = $\displaystyle [b_{0}D^2+ b_{1}D+b_{0}]y$

Therefore, $\displaystyle F= D[b_{0}D+ b_{1}+(b_{1}-b_{0})]y$.

I have that $\displaystyle b_{1}$ that I'm not sure how to get rid of.

Thank you for your time.