Differential Equations Workbook For Dummies®

Chapter 6. Handling Homogeneous Linear Higher Order Differential Equations

In This Chapter

• Reviewing the higher order process with real and distinct roots

• Adding complexity with complex roots

• Avoiding double duty with duplicate roots

This chapter is where you get to practice tackling higher order differential equations, where n > 2. (Note: Higher order equations are sometimes referred to as nth order equations.) A general linear higher order differential equation looks like this:

NOTE

Solving higher order differential equations where n = 3 or more is a lot like solving differential equations of the first or second order, with two exceptions: You need more integrations, and you have to solve larger systems of simultaneous equations to meet the initial conditions.

Every linear higher order differential equation you encounter in this chapter has constant coefficients, and the main way you can plan to tackle these problems is by attempting a solution of the form

y = e^rx

Substituting in this attempted solution results in a characteristic equation in powers of r, just as it does for the linear second order differential equations covered in Chapters 4 and 5. The problem here is that you're dealing with cubic (or higher!) characteristic equations, as well as 3 × 3 systems of simultaneous equations to handle the initial conditions.

Whenever you're facing a characteristic equation that's ...