Differential equations MOC

Reduction of order (homogenous second-order differential equation)

Reduction of order is a technique for finding the general solution of a homogenous second order linear DE¹ when a particular 𝑦1𝑥 solution is known.

We begin by assuming that 𝑦(𝑥) =𝑢(𝑥)𝑦1(𝑥) for some function 𝑢(𝑥) to be determined. It follows from this that

Given that 𝑦1 is indeed a solution, this substitution will reduce the DE to a first order separable DE on the independent variable 𝑢′ (see reasoning below), which can be then used to determine the general solution.

Explanation

We write the DE as 𝐿[𝑦] =0, where the linear operation 𝐿 is defined by

We are given that 𝐿[𝑦1] =0. Then

where 𝐾 is the second order linear operator

Method

In general, I have found it most effective to only substitute the particular solution 𝑦1 after the gathering different 𝑢 terms. Consider the ODE²

with particular solution 𝑦1 =𝑥−1/2sin⁡𝑥, so

We assume 𝑦 =𝑢𝑦1 is a solution to the ODE, thus

which is a separable ODE of order 2.

Practice problems

• 2017. Elementary differential equations and boundary value problems, p. 133 (§3.4 problems 18–22)

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