## Abel's Identity

Given a homogeneous linear Second-Order Ordinary Differential Equation,

 (1)

call the two linearly independent solutions and . Then
 (2)

 (3)

Now, take (3) minus (2),

 (4)

 (5)

 (6)

Now, use the definition of the Wronskian and take its Derivative,
 (7) (8)

Plugging and into (6) gives
 (9)

This can be rearranged to yield
 (10)

which can then be directly integrated to
 (11)

where is the Natural Logarithm. Exponentiation then yields Abel's identity
 (12)

where is a constant of integration.