Sturm-Liouville Applications

Example: Find the eigenvalues and eigenfunctions for the following problem:

y’’ + λ y = 0 here p(x) = 1, q(x) = 0, and r(x) = 1

hy(0) - y ’(0) = 0 , h > 0 and y(L) = 0

Note: Since both p(x) and r(x) > 0 there are no negative eigenvalues. So the only

Possible eigenvalues are λ = 0 and λ > 0 . The table below contains these two

cases.

Case 1: λ = 0 y(x) = Ax + B , y ’ = A

hy(0) - y ’(0) = 0 = hB - A = 0 , A = hB

y(L) = 0 = AL + B = hB + B = 0 = (h + 1)B = 0

Since h > 0, B = 0 and A = hB = 0

So there are no eigenvalues nor eigenfunctions for λ = 0 .

Case 2: λ > 0 , λ = α² y’’ + α² y = 0

y = A cos αx + B sin αx and y’ = ˗ Aα sin αx + Bα cos αx

hy(0) ˗ y’(0) = 0 = hA ˗ Bα , B = hA/α

So y = A cos αx + (hA/α) sin αx = A [cos αx + (h/α)sin αx] --------- (1)

and y(L) = 0 = A [cos αL + (h/α)sin αL] for a nontrivial solution A ≠ 0

Let β_n = α_nL so α_n = β_n/L --------------------------- (2)

So cos β_n + (hL/ β_n)sin β_n] = 0

tan β_n = ˗ β_n /hL (transcendental equation for roots)

where the eigenvalues are λ_n = α_n² = (β_n/L)²

and from (1) with (2) the eigenfunctions are y_n(x) = β_n cos β_nx/L + hLsin β_nx/L

where β_n is the nth positive root of tan x = ˗ x/hL (hint: use a plot)

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