Stepping beyond first-order problems, we turn to equations involving higher derivatives—especially the important class of linear equations with constant coefficients. This chapter explains why these equations are so useful and how systematic methods yield powerful, general solutions.
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.
An \(\DLO \text{LHCC}\) equation is a \(\DLO \text{L}\)inear \(\DLO \text{H}\)omogeneous \(\DLO \text{C}\)onstant \(\DLO \text{C}\)oefficient equation.
The derivative properties of exponential functions make them natural solutions for LHCC equations, and substituting \(y = e^{rx}\) into an LHCC equation leads to a polynomial equation in \(r\) called the characteristic equation.
If the characteristic polynomial factors easily, use algebraic techniques like grouping, factoring by common terms, or recognizing patterns such as difference of squares or cubes.
