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# Abel's Differential Equation 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)

Now, use the definition of the Wronskian and take its derivative,

 (5) (6) (7)

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.

Abel's Differential Equation, Second-Order Ordinary Differential Equation, Second-Order 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 Wolfram|Alpha

Abel's Differential Equation Identity

## Cite this as:

Weisstein, Eric W. "Abel's Differential Equation Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AbelsDifferentialEquationIdentity.html