The task of representing a differentiable functions as a polynomial is done with a Taylor expansion. We have looked at a few examples, and at the Maclaurin series as special case: for x=0.
Solving differential equations is an essential skill for any physicist. After classifying the differential equation, you know several methods of solving a first-order differential equation: separation of variables and integrating factor method. You also know, which solutions arise from homogeneous and inhomogeneous second-order differential equations with constant coefficients.

The following are from MathCentre on Differential Equations, and are used in this course with permission.