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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 r(d) Recall that the minimum value of the Rayleigh quotient for all continous functions u() satisfying the associated boundary

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-

Answers

Angư Geven G-Cx) are reel and Centmay ar) a, b) and p and Cx) are 8+xu oi d Tdx c6.2 to be saty-ffect. λ 20

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