(a) Show that a differential equation of the form a(x) u"+b(x) u' = /(x) is in self-adjoint form (11.12) if and only if b(x) = a'(x).
(b) If b(x) ≠ a'(x) and a(x) 0 everywhere, show that you can multiply the differential equation by a suitable integrating factor p(x) so that the resulting ordinary differential equation p(x) a(x)u" + p(x) b{x) u' = p(x) fix) is in self-adjoint form.
(c) Place the following differential equations in self-adjoint form, using an integrating factor if required:
(i) -x2u"- 2xn' = x - 1
(ii) e"u" + e'u' = e2x
(iii) u." + 2n' = 1
(iv) -xu" - 3u'=x
(v) cosx u" + sinx u' = cosx