Motion under a time-dependent driving force plus a linear velocity-dependent friction: From the unilateral Fourier transform to the Green function

The problem of a particle subject to a time-dependent driving force plus a linear velocity-dependent friction can be addressed by utilizing the unilateral Fourier transform, despite the presence of derivatives of odd and even order in the differential equation. This technique yields a system of algebraic equations that combine the Fourier sine and cosine transforms. While this method is useless for solving the homogeneous equation, it can be effectively used to obtain an integral representation of the particular solution. Remarkably, this integral representation of the particular solution is expressed in terms of the Green function. This type of exactly solvable problem is relevant for students who are studying mathematical methods applied in the fields of physics and engineering at the undergraduate level, as it can serve as a useful illustration of how unilateral Fourier transforms can be employed to solve problems and to develop an understanding of Green functions, even in introductory calculus courses.