How to reduce differential operator to a 4th order ODE of the standard Euler type?

Question

I read this article https://mae.ufl.edu/~uhk/STOKES-DRAG-FORMULA.pdf How do we actually arrive to a 4th order ODE below?

Answer 1

Why don't you just plug in? Observe $$ D\sin^2 \theta \equiv \sin\theta ~\partial_\theta\! \left (\frac {1}{\sin\theta}\partial_\theta \right )~ \sin^2 \theta= -2 \sin^2 \theta, $$ so that you may substitute the eigenvalue for D, $$ Q^2 f \sin^2\theta = \left (\partial_r^2 +{D\over r^2}\right)^2 f \sin^2\theta= \sin^2\theta ~~ (\partial_r^2-2/r^2)^2f\\ = \sin^2\theta ~~\left (f''''-{4\over r^2} f''+{8\over r^3} f' -{8\over r^4}f\right). $$