Linear Second-Order Differential Equations Part 1: Homogeneous Case

After a number of tutorials covering first-order differential equations, it's time to start tackling second-order differential equations. These contain a second derivative term, and they are quite useful in physics. To introduce these, let's first examine the simplest case, a linear homogenous second-order differential equation that exhibits constant coefficients. To solve these we will need to find a characteristic equation and do some algebra, a process which will vary depending on whether we are dealing with real distinct roots, complex roots, or a repeated root. Let's take a look!