6. Consider Bessel's equation, x'u" +ru' t (22 - a? )u =0, which, when a is not an integer, has the two solutions, u1(x) = Ja(x) and u2(x) = J-a(x), where Ja(x) is called the Bessel function of the first kind of order o. The general solution may then be written as u(x) = cJa(x) +c2J-a(x). Bessel's equation comes from solving some PDEs, such as the heat or wave equations, in cylindrical coordinates. Now consider the spherical Bessel equation, ray" + 2xy' + (x2 - n(n + 1))y =0, where n is an integer. This comes from solving some PDEs in spherical coordinates instead. Use the substitution y = r u to transform the spherical Bessel equation for y into Bessel's equation for u. (a) Differentiate the substitution y = r zu to write y' and y" in terms of r, u, u', and u". (b) Plug these expressions for y, y', and y" into the spherical Bessel equation to find Bessel's equation for u. What is the order of the equation, o? Is o an integer? Hint: Distribute and refactor the quadratic in n. (c) Write out the general solution, u(r), in terms of the Bessel functions to find the general solution of the spherical Bessel equation, y(r)