proposition 9.38 Liouville's formula

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1347 · p. 287

Rests on

depends_on definition 9.37 Fundamental matrix ¶
- depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
  - depends_on definition 9.2 Linear equation; homogeneity ¶
    - depends_on definition 9.1 Ordinary differential equation ¶
  - depends_on remark 9.3 Normal form and first-order systems ¶
  - depends_on theorem 9.8 Picard–Lindelöf ¶
    - depends_on definition 9.4 Lipschitz condition ¶
    - depends_on equation 9.3 eq:slt-ode-system ¶
    - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on proposition 7.47 Comparison; absolute convergence ¶
      - proves proof ch:07-odes-sturm-liouville@proof-1 ¶
    - proves proof ch:07-odes-sturm-liouville@proof-2 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-3 ¶
- depends_on theorem 9.13 Dimension of the solution space ¶
  - depends_on corollary 9.9 Linear equations: existence on the whole interval ¶ ↺
  - depends_on definition 9.2 Linear equation; homogeneity ¶ ↺
  - proves proof ch:07-odes-sturm-liouville@proof-6 ¶
depends_on proposition 9.15 Abel's identity ¶
- depends_on definition 9.2 Linear equation; homogeneity ¶ ↺
- depends_on definition 9.14 Wronskian ¶
- proves proof ch:07-odes-sturm-liouville@proof-7 ¶
proves proof ch:07-odes-sturm-liouville@proof-20 ¶

Supports

depends_on corollary 9.41 Hill's equation ¶
- depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
depends_on theorem 9.39 Floquet ¶
- depends_on corollary 9.41 Hill's equation ¶ ↺

Neighborhood

Every logical edge within two steps of this node.

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partly declared
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Edges

type	direction	node	provenance	where
`depends_on`	→	Fundamental matrix	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1354
`depends_on`	→	Abel's identity	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1354
`depends_on`	←	Hill's equation	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1491
`depends_on`	←	Floquet	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1397
`proves`	←	ch:07-odes-sturm-liouville@proof-20	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1357