proposition 9.15 Abel's identity
open in the book ·
parts/02-mathematical-methods/07-odes-sturm-liouville.tex:519
· p. 278
Rests on
-
depends_on
definition 9.2
Linear equation; homogeneity
¶
- depends_on definition 9.1 Ordinary differential equation ¶
- depends_on definition 9.14 Wronskian ¶
- proves proof ch:07-odes-sturm-liouville@proof-7 ¶
Supports
-
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
corollary 9.41
Hill's equation
¶
- 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 equation; homogeneity | declared | parts/02-mathematical-methods/07-odes-sturm-liouville.tex:527 |
depends_on |
→ | Wronskian | declared | parts/02-mathematical-methods/07-odes-sturm-liouville.tex:527 |
depends_on |
← | Liouville's formula | declared | parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1354 |
depends_on |
← | Wronskian test for independence | declared | parts/02-mathematical-methods/07-odes-sturm-liouville.tex:554 |
proves |
← | ch:07-odes-sturm-liouville@proof-7 | declared | parts/02-mathematical-methods/07-odes-sturm-liouville.tex:530 |