# 3. (10 Points, part III) Consider the Sturm-Liouville differential equation where the coefficients p(z), q(z), and...

###### Question:

3. (10 Points, part III) Consider the Sturm-Liouville differential equation where the coefficients p(z), q(z), and σ(z) are real and continous on la, b , and p(2) and σ(z) are strictly positive for all a,b (a) Derive the Rayleigh quotient λ from (2). b) What does this quotient describe? Give two examples of applications for this formula. (c) what are the neces,ary conditions for λ > 0 to be satisfied?
(d) Recall that the minimum value of the Rayleigh quotient for all continous functions u() satisfying the associated boundary conditions (but not necessarily the differential equation) is the lowest eigenvalue: 1min In (2) let a-0 and b-1, p(z)-1, q(z)-0, and σ(z)-z? + and 0 < ơmin o(z) < ơmaz . Assume also homogenous Dirichlet boundary conditions. Determine an upper and a nonzero lower bound for the lowest eigenvalue λ1-

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