Question

Use the solutions obtained in Problem 3 to find the eigenvalues and normalized eigenfunctions of the Sturm-Liouville problem

Answer 1

Let us the consider equations rightarrow eigen from the data is rightarrow we can obtained by the source of problem 3 can be eigen values and the Normalized to eigen function of these are it is let us the data is (x'') + x = 0, x' (1) = 0 x(b) = 0 rightarrow let the first we can use for problem 3. are let the consider equations are x'' + lambda x = 0 and x' (0) = 0, x' (c) = 0 lambda = 0 and x'' = 0 and x = e_1 + e_2 x now they are two parts x' = e_2, x' (0) = e_2 = 0 and x(e) = e_1 = 0 Therefore lambda = 0 is not be eigen value \r\n Part A: Now let us consider the equations are lambda < 0, lambda = -a^2, a notequalto 0) Now x'' - a^2 x = 0 let us the solve from the problem Rightarrow x(x) = e_1 e^ax + e_2 e^- ax Rightarrow x' (x) = a (e_1 e^ax - e_2 e^- ax)