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Method of Reduction for Second Order Differential Equations

In a Nut Shell: The method of reduction provides an alternative method to solving

linear, second order differential equations provided one solution is known. Then use it

to reduce the order of the differential equation.

Strategy: Suppose y₁(x) is a solution to the original differential equation. Then

introduce a new dependent variable, v(x), and combine it with y₁(x) to obtain a second

solution, y(x), for the original differential equation:

y(x) = y₁(x) v(x)

Procedure: Introduce this y(x) into the differential equation to obtain a new one

involving the new dependent variable, v(x). As an intermediate check on your

procedure, this substitution should result in a differential equation with the derivatives

of v(x), that with the introduction of a second dependent variable, say w(x), can be

integrated or separated leading to a differential equation of lower order. Once you

arrive at this lower order differential equation in terms of derivatives of w(x), you

then integrate to find v(x). Finally combine v(x) with y₁(x) to obtain the general

solution of the original differential equation.