By Carl M. Bender

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**Advanced mathematical methods for scientists and engineers**

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9) yiP) = 0, can always be transformed to Sturm-liouville d [ p(x)dY ] + [q(x) + E[(x)]y = 0, -d x dx y(a) = y(/3) = o. 10) is a Sturm-Liouville problem. It is a mathc ,Iatical property of Sturm-Liouville problems which we do not prove here that when p(x) > 0, q(x) 5: 0, r(x) > 0, a::; x $; /3, there are an infinite number of eigenvalues E = Eo, E " E 2 , . which are all real and positive. Moreover, the eigenfunctions y. (11 = 0, 1,2, ... ) associated with the eigenvalues E. can be normalized so that they are onhonormal with respect to the weight Junelion r(x): if n ="', if oim.

The farmer still starts 20 yd north of the pig but can now run at a speed of tv. The farmer is assisted by his daughter who starts 15 yd south of the pig at t = 0 and can run at a speed of ~v. lfboth the fanner and the farmer's daughter always run toward the instantaneous position of the pig, who catches the pig first? (e) At [ = 0, a pig initially at (I, 0) starts to run around the unit circle with constant speed v. At [ 0, a farmer initially at the origin runs with constant speed v and instantaneous velocity directed toward the instantaneous position of the pig.

3) Again, y(x) 0 is a trivial solution for all E; the eigenvalues are determined by the condition that there exist nontrivial solutions. Since the general solution of y" + Ey = 0 that satisfies y(O) = 0 is y(x) A sin (xft1 y'(l) = 0 requires that AJE cos = O. Therefore, the eigenvalues are ft E = (1n)l, (in)', Gn)l, Example 3 Eigenvalue problem on an infinite domain. The eigenvalue problem y" + (E - tx')y 0, y(± (0) O. '«00, (1Mb) is known as the quantum harmonic oscillator. For any E this problem has the trivial solution y(x) =0 0, but for special £ there are nontrivial solutions.