Abel's Theorem
For with continuous , the Wronskian satisfies , so it is either zero or never zero on an interval.
Given any 2 solutions of
if and are continuous on an interval , then either the Wronskian for all or for all .
Proof sketch
Write . Differentiating and using for each solution,
This is a separable ODE for :
where the exponential factor is never zero. This is Abel’s identity. Hence if , then for all ; otherwise for all .
Geometrically, the solution vectors are either always collinear or never collinear in the phase space.
Consequences and applications
If , then , so the Wronskian is constant.
Abel’s identity also finds without knowing the solutions explicitly. For Bessel’s equation , rewritten as ,
The identity extends to solutions of th order homogeneous linear ODEs.
Given one solution , Abel’s identity gives a second: from
dividing both sides by ,
Related
Stated in
- Theorem 6.12 (Abel's Theorem)§6.3.4 Abel’s Theorem