theorem 9.13 Dimension of the solution space

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:480 · p. 278

Rests on

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 definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on proposition 7.47 Comparison; absolute convergence ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - depends_on definition 7.45 Series ¶
      - depends_on theorem 7.8 Cauchy criterion ¶
      - proves proof ch:05-real-analysis@proof-28 ¶
    - 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 definition 9.2 Linear equation; homogeneity ¶ ↺
proves proof ch:07-odes-sturm-liouville@proof-6 ¶

Supports

depends_on definition 9.37 Fundamental matrix ¶
- depends_on proposition 9.38 Liouville's formula ¶
  - 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 ¶ ↺
- depends_on theorem 9.39 Floquet ¶ ↺
depends_on proposition 28.18 The three regimes ¶
depends_on proposition 28.4 General solution and isochronism ¶
- depends_on phenomenon 28.9 Isochronism of the small-amplitude pendulum ¶
  - assumes experiment Kater: the reversible pendulum ¶
  - tests experiment Galileo and Huygens: isochronism ¶
depends_on proposition 9.106 The modified series and what it does not do ¶
depends_on proposition 9.28 The scalar constant-coefficient equation ¶
- depends_on proposition 9.88 The model problem, and the error of the composite ¶
depends_on proposition 9.16 Wronskian test for independence ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	Linear equations: existence on the whole interval	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:486
`depends_on`	→	Linear equation; homogeneity	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:486
`depends_on`	←	Fundamental matrix	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1344
`depends_on`	←	The three regimes	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:538
`depends_on`	←	General solution and isochronism	declared	parts/03-classical-mechanics/11-oscillations-waves.tex:113
`depends_on`	←	The modified series and what it does not do	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:4122
`depends_on`	←	The scalar constant-coefficient equation	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1024
`depends_on`	←	Wronskian test for independence	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:554
`proves`	←	ch:07-odes-sturm-liouville@proof-6	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:489