Definition:Conjugate Point/Definition 1
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Definition
Let:
- $-\map {\dfrac \d {\d x} } {P h'} + Q h = 0$
with boundary conditions:
- $\map h a = 0, \quad \map h c = 0, \quad a < c \le b$
Suppose:
- $\map h x = 0 \quad \neg \forall x \in \closedint a b$
Suppose:
- $\map h a = 0, \quad \map h {\tilde a} = 0, \quad a \ne \tilde a$
Then the point $\tilde a$ is called conjugate to the point $a$ with respect to solution to the aforementioned differential equation.
Are both a and c conjugate?
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Sources
As such, the following source works, along with any process flow, will need to be reviewed.
Specifically: with respect to definitions 1, 2 and 3
If you have access to any of these works, then you are invited to review this list, and make any necessary corrections.
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- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 5.26$: Analysis of the Quadratic Functional $ \int_a^b \paren {P h'^2 + Q h^2} \rd x$