Second-Order Ordinary Differential Equation Second Solution

If one solution () to a second-order ordinary differential equation

(1)

is known, the other () may be found using the so-called reduction of order method. From Abel's differential equation identity

(2)

where

(3)

is the Wronskian.

Integrating gives

(4)

(5)

and solving for  gives

(6)

But

(7)

so combining (◇) and (◇) yields

(8)

(9)

Disregarding , since it is simply a multiplicative constant, and the constants  and , which will contribute a solution which is not linearly independent of , leaves

(10)

In the special case , this simplifies to

(11)

If both general solutions to a second-order nonhomogeneous differential equation are known, variation of parameters can be used to find the particular solution.