# CHAPTER 3

# Higher Order Linear ODEs

The concepts and methods of solving linear ODEs of order *n* = 2 extend nicely to linear ODEs of higher order *n*, that is, *n* = 3, 4, etc. This shows that the theory explained in Chap. 2 for second-order linear ODEs is attractive, since it can be extended in a straightforward way to arbitrary *n*. We do so in this chapter and notice that the formulas become more involved, the variety of roots of the characteristic equation (in Sec. 3.2) becomes much larger with increasing *n*, and the Wronskian plays a more prominent role.

The concepts and methods of solving second-order linear ODEs extend readily to linear ODEs of higher order.

This chapter follows Chap. 2 naturally, since the results of Chap. 2 can be readily extended to that of Chap. 3.

*Prerequisite:* Secs. 2.1, 2.2, 2.6, 2.7, 2.10.

*References and Answers to Problems:* App. 1 Part A, and App. 2.

## 3.1 Homogeneous Linear ODEs

Recall from Sec. 1.1 that an ODE is of *n*th order if the *n*th derivative *y*^{(n)} = *d*^{n}y/dx^{n} of the unknown function *y*(*x*) is the highest occurring derivative. Thus the ODE is of the form

where lower order derivatives and *y* itself may or may not occur. Such an ODE is called **linear** if it can be written

(For *n* = 2 this is (1) in Sec. 2.1 with *p*_{1} = *p* and *p*_{0} = *q*.) The **coefficients** *p*_{0}, …, *p*_{n−1}