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 π¦1π₯ solution is known.
We begin by assuming that π¦(π₯)=π’(π₯)π¦1(π₯) for some function π’(π₯) to be determined. It follows from this that
Given that π¦1 is indeed a solution, this substitution will reduce the DE to a first order separable DE on the independent variable π’β² (see reasoning below),
which can be then used to determine the general solution.
Explanation
We write the DE as πΏ[π¦]=0, where the linear operation πΏ is defined by
In general, I have found it most effective to only substitute the particular solution π¦1 after the gathering different π’ terms.
Consider the ODE2
π₯2π¦β³+π₯π¦β²+(π₯2β14)π¦=0
with particular solution π¦1=π₯β1/2sinβ‘π₯, so