Laplacian in curvilinear coordinates
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The Laplacian nabla^2 in Cartesian coordinates is the simple sum u_xx + u_yy + u_zz, but most real problems do not come in boxes. A planet is round, a pipe is a cylinder, an atom is roughly spherical, a drumhead is a disk. To use separation of variables you must align the coordinate system with the boundary, and that means rewriting the Laplacian in polar, cylindrical, or spherical coordinates — where it suddenly grows extra terms that account for the curving of the coordinate lines.
In plane polar coordinates (r, theta), the Laplacian becomes nabla^2 u = u_rr + (1/r) u_r + (1/r^2) u_theta theta. The extra (1/r) u_r and the 1/r^2 factor are not arbitrary — they are the price of the radial lines fanning out, so that equal steps in angle cover more ground far from the origin. Cylindrical coordinates (r, theta, z) just add a flat u_zz to the polar form. Spherical coordinates (r, theta, phi) give the more elaborate nabla^2 u = (1/r^2)(d/dr)(r^2 u_r) + (1/(r^2 sin theta))(d/d theta)(sin theta u_theta) + (1/(r^2 sin^2 theta)) u_phi phi. These are special cases of one general formula built from the scale factors of the coordinate system (how far you move in space per unit change of each coordinate). When you separate variables in these coordinates, the radial part is no longer a simple sine or cosine: the 1/r and 1/r^2 terms turn the radial ODE into Bessel's equation in the cylinder and into Legendre's equation (in the angle) in the sphere.
This is precisely the gateway from elementary PDEs to the special functions of mathematical physics. Solving Laplace's, the heat, or the wave equation on a disk or cylinder forces Bessel functions upon you; solving them on a sphere forces Legendre polynomials and spherical harmonics. The hydrogen atom, the modes of a circular drum, the multipole expansion of a charge distribution, antenna radiation patterns — all of these inherit their characteristic functions directly from writing the Laplacian in the coordinate system that matches the geometry.
On a circular drum, Laplace's equation in polar coordinates separates as u = R(r) Theta(theta); the angular part gives cos(m theta) and sin(m theta), and the radial part R(r) satisfies Bessel's equation, so the drum's vibration modes are Bessel functions J_m times trigonometric functions of the angle.
Matching the coordinates to a round boundary turns the radial ODE into Bessel's equation — special functions enter through the geometry.
A frequent slip is to forget the first-derivative terms — the (1/r) u_r in polar or the (2/r) u_r in spherical: dropping them is not a minor approximation but a different equation entirely, and it is exactly those terms that generate Bessel and Legendre functions.