definition 9.2 Linear equation; homogeneity

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:134 · p. 274

ground object -- no derivation owed

Rests on

depends_on definition 9.1 Ordinary differential equation ¶

Supports

depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
- 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 lemma A.439 Existence, uniqueness, and the structure of the zeros ¶
  - depends_on lemma A.442 The Wronskian of the accessory equation is constant ¶
    - depends_on definition A.467 Green function of the shifted problem ¶
      - depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
      - depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
    - depends_on theorem A.443 Sturm separation theorem ¶
  - depends_on theorem A.441 Continuous dependence on the data and on the initial point ¶
    - depends_on corollary A.445 Displacing the initial point ¶
- depends_on proposition 20.9 Linear resistance: exact motion, and the lost range ¶
  - depends_on remark 20.11 What has no closed form, and what to do about it ¶
- depends_on proposition 9.103 The positive zeros and their spacing ¶
- depends_on proposition 9.18 The linear equation: integrating factor ¶
  - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
    - depends_on proposition 9.88 The model problem, and the error of the composite ¶
- depends_on theorem A.74 Flow of a time-dependent vector field ¶
  - depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
    - depends_on lemma A.548 The joint flow is a translation action on the level set ¶
      - depends_on corollary A.553 Quasi-periodic motion ¶
      - depends_on lemma A.549 The action is transitive ¶
      - depends_on proposition A.552 The component is a torus ¶
    - depends_on lemma A.567 The differential of the momentum map along the orbit ¶
      - depends_on theorem A.568 The radical is the isotropy orbit ¶
    - depends_on proposition 13.131 Commuting fields have commuting flows ¶
      - depends_on lemma A.548 The joint flow is a translation action on the level set ¶ ↺
      - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
      - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶
    - depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶ ↺
    - depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶ ↺
- depends_on theorem 9.23 Solution of a constant-coefficient system ¶
  - depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
    - depends_on corollary 9.27 Linear stability ¶
      - depends_on corollary 9.41 Hill's equation ¶ ↺
      - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
      - depends_on proposition 9.29 The planar classification ¶
      - depends_on theorem 9.39 Floquet ¶ ↺
    - depends_on proposition 9.29 The planar classification ¶ ↺
  - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶ ↺
  - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶ ↺
  - depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶ ↺
  - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
    - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
      - depends_on definition 32.22 Bifurcation ¶
      - depends_on example 32.18 A linear centre that is really a stable focus ¶
  - depends_on theorem 9.26 Structure of the solutions ¶
    - depends_on corollary 9.27 Linear stability ¶ ↺
- depends_on theorem 9.13 Dimension of the solution space ¶
  - depends_on definition 9.37 Fundamental matrix ¶ ↺
  - 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 ¶
depends_on proposition 30.57 Bending waves are dispersive ¶
- depends_on example 30.58 A steel ruler ¶
- depends_on remark 30.59 Where Euler–Bernoulli fails ¶
depends_on proposition 9.15 Abel's identity ¶
- depends_on proposition 9.38 Liouville's formula ¶ ↺
- depends_on proposition 9.16 Wronskian test for independence ¶ ↺
depends_on theorem 9.13 Dimension of the solution space ¶ ↺

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`	→	Ordinary differential equation	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:146
`depends_on`	←	Linear equations: existence on the whole interval	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:364
`depends_on`	←	Bending waves are dispersive	declared	parts/03-classical-mechanics/13-continuum-elasticity.tex:2103
`depends_on`	←	Abel's identity	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:527
`depends_on`	←	Dimension of the solution space	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:486