How do we solve a second order, homogeneous, linear differential equation?

First of all let's think about what this really means:

Homogeneous means that all term in the differential equation contain the dependent variable(in most cases y). The dependent variable is the variable that varies with another variable, known as the independent variable.(An easy way to spot it - the dependent variable is always being differentiated, and it is never the domain in which differentiation occurs)

Linear means that the dependent variable is not being multiplied by itself, and is not contained in any exponential, trigonometric of non-linear function. This simplifies differential equations.

Second order defines the largest derivative contained in the equation. Hence this equation must contain the term f''(x).

Solving this kind of equation is easily done by using a neat trick. If you let y=Ae^mx, then y'=mAe^mx and y''=m²Ae^mx. Substituting these values in results in a linear quadratic equation, which is fairly straightforward. Depending on what values for m we get, there are different ways of treating this differential equation.