A review of the problem of motion for masses attached to springs.That's the physics of the chapter, a familiar problem from high school and introductory college classes, meant to orient and refresh the reader.The mathematical lesson is about series solutions (the method of Frobenius) for ordinary differential equations (ODEs), and the definition of trigonometric special functions in terms of the ODEs that they solve.This is the chapter that reviews complex numbers and the basic properties of ODEs and their solutions (superposition, continuity, separation of variables).