Problem Description:

Find the general solution to the following equation using the general form of Bessel's equation:

Problem Solution:

By equating the coefficients of the y’(x) term with the most general form of Bessel’s equation [see eqns. (8.59) - (8.61)], we have

Therefore, a = -3, b = 2, and p = 4.

With these constants specified, the coefficient for the y(x) term becomes

Therefore, c = 3, d = -5, and q = 1.

Now, since the conditions of the method are satisfied, the constants within the general solution become

Finally, since d < 0, we have

This represents an analytical solution to the given problem (a tough problem indeed). In a realistic BVP, one would now apply appropriate boundary conditions to uniquely identify the two arbitrary coefficients within the general solution.