EDB — 1QB

↑ ← → ↓

View
Tools

View

English

23 Ordinary Differential equations[1QB]

To solve the following exercises, it is imporant to know some fundamental results, such as: the existence and uniqueness theorem ¹ , Gronwall’s Lemma; and in general some methods to analyze, solve and qualitative study Ordinary Differential Equations (abbreviated ODE). These may be found e.g. in [ 29 , 24 , 3 ] .

E4: [1QC]
E4: [1QH]
E4: [1QK]
E4: [1QN]

23.1 Autonomous problems

E4: [1QR]
E4: [1QV]
E4: [1QX]
E4: [1QZ]

23.2 Resolution

E4: [1R1]
E4: [1R4]

23.3 Qualitative discussions

[ [1R6]]

For the following exercises the following simple comparison lemma may be useful.

Lemma 5

[1R7]

(There are much more refined versions of this lemma, see for example in section 8.6 in the course notes [ 3 ] ).

E5: [1R8]
E5: [1RD]
E5: [1RK]
E5: [1RQ]

QuasiEsercizio 1

[1RR]

QuasiEsercizio 2

[1RS]

23.4 Envelope

Given a family of planar curves, we want to define the envelope curve. Let’s see two possible definitions.

Definition 6 Curve Envelope

[23Y]

Remark 7

[240]

[ [1RT]]

We want to see that the two previous definitions are equivalent in this sense.

E7: [1RV]
E7: [1RY]
E7: [1S1]
E7: [1S4]
E7: [1S7]
E7: [1S9]

23.5 Linear equations (with constant coefficients)

Definition 8

[23Z]

E8: [1SC]
E8: [1SD]
E8: [1SF]
E8: [1SH]
E8: [1SK]
E8: [1SN]
E8: [1SP]
E8: [1SS]

23.6 Matrix equations

To solve the following exercises you need to know the elementary properties of the exponential of matrices, see section [2D8].

E8: [1SW]
E8: [1SY]
E8: [1T1]
E8: [1T3]
E8: [1T6]

Download PDF

Authors: "Mennucci , Andrea C. G." .

Bibliography

[29] Gerald Teschl. Ordinary differential equations and dynamical systems, volume 140. American Mathematical Soc., 2012. ISBN 978-0-8218-8328-0. URL http://www.mat.univie.ac.at/~gerald/ftp/book-ode/index.html. (Freely available on the author’s website).
[24] to3em. Ordinary Differential Equations in Rn. Springer, 1984. ISBN 978-0-387-90723-9. DOI: 10.1007/978-1-4612-5188-0.
[3] L. Ambrosio, C. Mantegazza, and F. Ricci. Complementi di matematica. Scuola Normale Superiore, 2021. ISBN 9788876426933. URL https://books.google.it/books?id=1QR0zgEACAAJ.
[23] Livio C. Piccinini, Giovanni Vidossich, and Guido Stampacchia. Equazioni differenziali ordinarie in \(R^ N\) (problemi e metodi). Liguori Editore, 1978.

Book index

ODE
Picard
Lindelöf
Cauchy
Lipschitz
theorem, existence and uniqueness —
theorem, Picard–Lindelöf —
theorem, Cauchy–Lipschitz —
ordinary differential equation , see ODE
differential equation , see ODE
Gronwall
lemma, Gronwall's —

Managing blob in: Multiple languages

This content is available in: Italian English