Advanced Engineering Mathematics, 7th Edition

3.2 Reduction of Order

INTRODUCTION

In Section 3.1 we saw that the general solution of a homogeneous linear second-order differential equation

a₂(x)y″ + a₁(x)y′ + a₀(x)y = 0 (1)

was a linear combination y = c₁y₁ + c₂y₂, where y₁ and y₂ are solutions that constitute a linearly independent set on some interval I. Beginning in the next section we examine a method for determining these solutions when the coefficients of the DE in (1) are constants. This method, which is a straightforward exercise in algebra, breaks down in a few cases and yields only a single solution y₁ of the DE. It turns out that we can construct a second solution y₂ of a homogeneous equation (1) (even when the coefficients in (1) are variable) provided that we know one nontrivial ...