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 nth order if the nth derivative y(n) = dny/dxn 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 p1 = p and p0 = q.) The coefficients p0, …, pn−1

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