Theorem: Let $a, b \in \mathbb{R} $ and let: $p_n(x)y_{x+n}+...+p_0(x)y_{x}=r(x) $ be a LDE on [a,b] with for all $x \in [a,b]$, $p_n(x)p_0(x) \neq 0$. let $x_0 \in \mathbb{R}^{[a,b+n]} $ be given. Now there is a unique solution $\phi \in \mathbb{R}^{[a,b+n]} $ to our equation on [a,b] such that for all $x \in [a,a+n)$ : $\phi(x)=x_0(x)$.
I don't understand this theorem, I want someone explain it with one example (can be 2nd order LDE) and how to find $\phi$ or $\phi_1$,$\phi_2$,...$\in \mathbb{R}^{[a,b+n]}$ (canonical solutions!?) and how to apply this theorem?
My understandings:
$x_0$ is initial condition. If $p_i(x)$ are constant. I know how to find set of fundamental solutions by taking $y=a^x$ , find characteristics equations and solve for a's and then construct set of fundamentals for Homogeneous LDE. I kind of understand $\mathbb{R}^{[a,b+n]}$ is a map operator ( but dont understand why it is helpful here as well?
also what and how to apply this theorem if $p_i(x) $ are not constant but periodic constants.
In case there is a book explaining this theorem please provide reference. because our Prof. dont provide any ref.