Up to this point, we’ve focused mostly on first-order equations. Now it’s time to take the next step: equations involving second derivatives and beyond. Among these, one class stands out for its importance and simplicity—the linear homogeneous differential equations with constant coefficients (often shortened to LHCC equations).
In this chapter, we’ll pin down exactly what those words mean: linear, homogeneous, and constant coefficient. Then we’ll discover why exponential functions are the natural “building blocks” for their solutions, and how the characteristic equation turns a differential equation into an algebra problem we can solve systematically.
By the end, you’ll have a clear strategy for solving LHCC equations of any order—a foundation that will carry directly into the next chapter, where we’ll look at what happens when these equations are no longer homogeneous.
