Given a homogeneous linear secondorder
ordinary differential equation,

(1)

call the two linearly independent solutions and . Then

(2)


(3)

Now, take (3) minus (2),

(4)

Now, use the definition of the Wronskian and take its
derivative,
Plugging and into (4) gives

(8)

This can be rearranged to yield

(9)

which can then be directly integrated to

(10)

where is the natural logarithm.
Exponentiating then yields Abel's identity

(11)

where is a constant of integration.
See also
Abel's Differential Equation,
SecondOrder Ordinary Differential
Equation,
SecondOrder
Ordinary Differential Equation Second Solution
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References
Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley,
pp. 118, 262, 277, and 355, 1986.Referenced on WolframAlpha
Abel's Differential
Equation Identity
Cite this as:
Weisstein, Eric W. "Abel's Differential Equation Identity." From MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/AbelsDifferentialEquationIdentity.html
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