Given a homogeneous linear second-order ordinary differential equation,
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(1)
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call the two linearly independent solutions 
and 
. Then
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(2)
|
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(3)
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(
(![y_1[y_2^('')+P(x)y_2^'+Q(x)y_2]-y_2[y_1^('')+P(x)y_1^'+Q(x)y_1]=0
(y_1y_2^('')-y_2y_1^(''))+P(y_1y_2^'-y_1^'y_2)+Q(y_1y_2-y_1y_2)=0
(y_1y_2^('')-y_2y_1^(''))+P(y_1y_2^'-y_1^'y_2)=0.](/images/equations/AbelsDifferentialEquationIdentity/NumberedEquation4.svg) |
(4)
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Now, use the definition of the Wronskian and take its derivative,
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(5)
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(6)
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(7)
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Plugging 
and 
into (4) gives
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(8)
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This can be rearranged to yield
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(9)
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which can then be directly integrated to
![ln[(W(x))/(W_0)]=-intP(x)dx,](/images/equations/AbelsDifferentialEquationIdentity/NumberedEquation7.svg) |
(10)
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where 
is the natural logarithm.
Exponentiating then yields Abel's identity
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(11)
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where 
is a constant of integration.
