Range of differential operator

Question

Show that the range of the operator $L$ is the whole space $C[a,b]$, and hence the inverse $L^{-1}$ has domain C[a,b].

$L:u \rightarrow -u''+p(x)u'+q(x)u$

$u \in dom(L)= \{u\in C^{2}[a,b], u(a)=0, u'(a)=0 \} $

Answer 1

Showing that $\text{ran} L=C[a,b]$ is equivalent to showing that, for each $f\in C[a,b]$, $Lu=f$ has a solution $u$ satisfying $u(a)=u'(a)=0$. But $Lu=f$ is a second order linear DE which has a unique solution $u$ for $\forall f\in C[a,b]$. Done.