Let $f\in C[0,1]$ be a continuous function and consider for $x\in(0,1)$ the Sturm-Liouvile problem $$ -u''(x)+x\cdot u(x)=f(x) \tag1$$ where $u'(0)=u'(1)=0.$
I need to show that for any $f\in C[0,1]$ there is a unique $u\in C^2[0,1]$ that satisfies (1).
Is there someone who knows a good book where I can find this result?