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PHYS 201 Mathematical Physics, Fall 2015, Homework 7 Due date: Thursday, December 3rd, 2015 1. Find the Frobenius series expansion about x = 0 of the solutions of the equation 2xy 00 − y 0 + x2 y = 0 2. This exercise follows Arfken 10.5.3(a), 10.5.4 to 10.5.7. i. Show that the Green’s function for the operator dy(x) d x Ly(x) = dx dx is ( − ln t, G(x, t) = − ln x, 0≤x<t t<x≤1 (1) Due to the pathology at the boundary point, the solution to the inhomogeneous equation involving L in terms of the Green’s function has an extra boundary term. ii. Show Green’s theorem in one dimension for a Sturm-Liouville type operator: Z b dv(t) d du(t) d p(t) − v(t) p(t) dt u(t) dt dt dt dt a b du(t) dv(t) − v(t)p(t) = u(t)p(t) dt dt a iii. Using Green’s theorem in the form above, let d dy(t) v(t) = y(t) and p(t) = −f (t) dt dt d ∂G(x, t) u(t) = G(x, t) and p(t) = −δ(x − t) dt ∂t and show that it yields t=b Z b dy(t) ∂ y(x) = G(x, t)f (t)dt + G(x, t)p(t) − y(t)p(t) G(x, t) dt ∂t a t=a iv. For p(t) = t, y(t) = −t and G(x, t) as in eq. 1, verify that the integrated part does not vanish. 1 3. Solve the following initial value problem for t > 0 in terms of Green’s functions: d2 y dy + α + βy = f (t), 2 dt dt y(0) = A, 4. Compute the inverse Laplace transform of f (s) = k2 s(s2 + k 2 ) i. By expanding in partial fractions, and ii. From the calculus of residues. 2 dy (0) = B dt