The solutions of linear first- and second-order ordinary differential equations can be unified based on the Taylor series expansion. Correspondingly the solutions of elementary linear second-order ordinary differential equations with constant coefficients can be related geometrically from a unified viewpoint. A common geometrical characteristic among different types of equations can be found, regardless of which form the Taylor series expansion takes. Exploring common characteristics as well as solving the differential equations is instructive to improve a student’s ability to think mathematically.

second-order ordinary differential equations with constant coefficients, Taylor series, hyperbolic function, circular function, quadratic function.