Chapter 12.16, Problem 13P

Verify by direct substitution that the text solution of equation (16.3) and your solutions in the problems above are correct. Also prove in general that the solution (16.2) given for (16.1) is correct. Hint: These are exercises in partial differentiation. To verify the solution (16.4) of (16.3), we would change variables from x, y to say z, u where $y=x^{1/2}u,u=J_{1/3}(z),z=2x^{3/2},$ and show that if x, y satisfy (16.3), then u, z satisfy (12.1), that is, $z^2\frac{d^{2}u}{d{z}^2}+z\frac{du}{dz}+\left(z^2-\frac{1}{9}\right)u=0.$