664 CHAPTER 33. BOUNDARY VALUE PROBLEMS, FOURIER SERIES

Then determine the equation solved by z. Hint: This is a little involved. First verifythat the left side reduces to

ddt

(p

ddx

((pq)−1/4

)z+(pq)1/4 dz

dt

)√qp+(λq+ r)(pq)−1/4 z = 0

Next verify that the z′ (t) terms all cancel. That way, in the above, you can neglectthese terms in using the product rule. This leads to(

ddx

(− 1

4 p−1/4q−5/4 ddx (pq)

)(p−1/4q3/4

) +r (pq)−1/4(p−1/4q3/4

))z+ z′′+λ z = 0

Now argue that the equation is of the form

z′′+(λ +m(t))z = 0

where m(t) is a function which depends on p,q.

25. Consider the eigenvalue problem for Bessel’s equation,

x2y′′+ xy′+(λx2−n2)y = 0, y(L) = 0

Show it can be written in self adjoint form as

(xy′)′+

(λx− n2

x

)y = 0

Thus in this case, q(x) = x and r (x) =−n2/x. What is the form of the equation if Li-ouville’s transformation is applied to this Bessel eigenvalue problem? Hint: Just usethe specific description of what was obtained above and that r (x) = −n2/x, p(x) =q(x) = x, and so t = x. You should get something like

z′′+λ z+(

1−4n2

4x2

)z = 0

26. In the above problem, let λ = 1 and let n = 1/2 and use to find the general solutionto the Bessel equation in which ν = 1/2. Show, using the above, that this generalsolution is of the form

C1x−1/2 cosx+C2x−1/2 sinx.

27. Show that the polynomial q(x) of degree n which minimizes∫ 1

−1| f (x)− p(x)|2 dx

out of all polynomials p of degree n is the nth partial sum of the Fourier seriestaken with respect to the Legendre polynomials q(x) = Sn f (x) , where Sn f (x) ≡∑

nk=0 ck pk (x) ,ck =

∫ 1−1 qk (x) f (x)dx .