Advanced Engineering Mathematics, 7th Edition

3.3 Linear Equations with Constant Coefficients

INTRODUCTION

We have seen that the linear first-order DE y′ + ay = 0, where a is a constant, possesses the exponential solution y = c₁e^−ax on the interval (−∞, ∞). Therefore, it is natural to ask whether exponential solutions exist for homogeneous linear higher-order DEs

a_ny⁽ⁿ⁾ + a_n−1y⁽ⁿ⁻¹⁾ + … + a₁y′ + a₀y = 0, (1)

where the coefficients a_i, i = 0, 1, …, n are real constants and a_n ≠ 0. The surprising fact is that all solutions of these higher-order equations are either exponential functions or are constructed out of exponential functions.

Auxiliary Equation

We begin by considering the special ...