Question: Consider the Sturm-Liouville problem

1. . Show that if \(\lambda\) is an eigenvalue and \(\phi\) the corresponding eigenfunction then
   \[\lambda \int_{0}^{1} r \phi^{2} d x=\int_{0}^{1}\left(p \phi^{\prime 2}+q \phi^{2}\right) d x+\frac{b_{1}}{b_{2}} p(1) \phi^{2}(1)-\frac{a_{1}}{a_{2}} p(0) \phi^{2}(0)\]
   provided that \(a_{2} b_{2} \neq 0 .\) How must this result be modified if \(a_{2}=0\) or \(b_{2}=0 ?\)
2. . Show that if \(q(x) \geq 0\) and if \(b_{1} / b_{2}\) and \(-a_{1} / a_{2}\) are nonnegative, then the eigenvalue \(\lambda\) is nonegative.
3. . Under the conditions of part(b), show that the eigenvalue \(\lambda\) is strictly positive unless \(q(x)=0\) for each \(x\) in \(0 \leq x \leq 1\) and also \(a_{1}=b_{1}=0\)

Price: $2.99

Solution: The downloadable solution consists of 3 pages
Deliverable: Word Document