Chapter 4 - Higher Order Differential Equations

Abstract

This chapter discusses higher order differential equations. It defines nth order ordinary linear differential equation and provides several examples of this equation and their solutions. The chapter also highlights two theorems that show that under reasonable conditions the nth order homogeneous equation has a fundamental set of n solutions. It then shows solutions of nth order linear homogeneous equations with constant coefficients with various examples. The general solutions of the nth order homogeneous linear differential equation with constant coefficients are determined by the solutions of its characteristic equation. The chapter defines characteristic equation of the nth order homogeneous linear differential equation. The command DSolve can be used to solve nth order linear homogeneous differential equations with constant coefficients as long as n is smaller than 5. In cases when the roots of the characteristic equation are symbolically complicated, approximations of the roots of the characteristic equation can be computed with the command NRoots. The chapter provides solutions of nonhomogeneous equations with constant coefficients by the annihilator method. The auxiliary equation of higher order Cauchy–Euler equations is defined in the same way and the solutions of higher order homogeneous Cauchy–Euler equations are determined in the same manner as solutions of the higher order homogeneous differential equations with constant coefficients. The chapter further analyzes the ordinary differential equations with nonconstant coefficients, such as exact second-order, autonomous, and equidimensional equations.