where did $a_n$ come frmo

Question

mimi_x3 · Answer

https://gyazo.com/883cde0aa8e6b9e5e25345e595c95ab2

mimi_x3 · Answer

I'm stuck on 1(ii) i don't understand where the y_n = a_n ... comes form https://gyazo.com/fcc668f6875ec3691a2545464e447355

anonymous · Answer

(͡° ͜ʖ ͡°)

mimi_x3 · Answer

Confusion arises because q(i) https://gyazo.com/31c86fc0a5fa60bbd071a50b780b8d82 wgere it does not have y = a_n....

thomas5267 · Answer

I guess $a_n$ is a "correction factor" to make sure the boundary conditions are met.

I think 1(ii) is the solution to 1(i).  1(i) asks under what p the differential equation$y''-py=0$ has non-trivial solution on [0,a] that satisfies each of the three boundary condition.  1(ii) tells you that there are an infinite amount of eigenvalues $p_n=-\dfrac{n^2\pi^2}{a^2}=-\kappa_n^2$, where $n\geq0$ satisfying this equation.  For n=0 and p=0, the solution is any constant c (since $y''=0$ and y has to satisfy the boundary condition.)  For n>0, the solution is given by $y=a_n\cos(\kappa_n x)$.