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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?

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    Hint: this question is probably related to the Airy function: http://en.wikipedia.org/wiki/Airy_function2012-06-11

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