Consider the bvp
y''+(lamda)y=0 y'(0)=0, y'(pi/6)=0
a)assume lamda=0 and find the solution
b)assume lamda> 0 and find the solution
c)assume lamda< 0 and find the solution
d) determine the eigenvalues and associated eigenfunctions corresponding the eigenvalues.

has the corresponding characteristic equation (CE)

a. If , then the CE has one root, , and so the general solution to the ODE is

Given that , and

it follows that , and so

b. If , then the CE has two complex roots, , and the general solution is

With the given boundary values, we have

where .

and the solution would be

so that we still get

c. If , then the CE has two real roots, , so that the general solution is

From the boundary conditions we get

from which it follows that , so again the solution is

d. We only get eigenvalues in the case when , as in part (b):

for which we get the corresponding eigenfunctions

d

step-by-step explanation:

d