Ordinary Differential Equations: Graduate Level Problems ...

[Pages:100]Ordinary Differential Equations: Graduate Level Problems and Solutions

Igor Yanovsky

1

Ordinary Differential Equations

Igor Yanovsky, 2005

2

Disclaimer: This handbook is intended to assist graduate students with qualifying examination preparation. Please be aware, however, that the handbook might contain, and almost certainly contains, typos as well as incorrect or inaccurate solutions. I can not be made responsible for any inaccuracies contained in this handbook.

Ordinary Differential Equations

Igor Yanovsky, 2005

3

Contents

1 Preliminaries

5

1.1 Gronwall Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Linear Systems

7

2.1 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Fundamental Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Distinct Eigenvalues or Diagonalizable . . . . . . . . . . . . . . . 7

2.2.2 Arbitrary Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Asymptotic Behavior of Solutions of Linear Systems with Constant Co-

efficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Variation of Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Classification of Critical Points . . . . . . . . . . . . . . . . . . . . . . . 12

2.5.1 Phase Portrait . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.7 Stability and Asymptotic Stability . . . . . . . . . . . . . . . . . . . . . 23

2.8 Conditional Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.9 Asymptotic Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.9.1 Levinson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Lyapunov's Second Method

27

3.1 Hamiltonian Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Lyapunov's Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.1 Stability (Autonomous Systems) . . . . . . . . . . . . . . . . . . 29

3.3 Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4 Invariant Sets and Stability . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5 Global Asymptotic Stability . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6 Stability (Non-autonomous Systems) . . . . . . . . . . . . . . . . . . . . 41

3.6.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Poincare-Bendixson Theory

42

5 Sturm-Liouville Theory

48

5.1 Sturm-Liouville Operator . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2 Existence and Uniqueness for Initial-Value Problems . . . . . . . . . . . 48

5.3 Existence of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.4 Series of Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.5 Lagrange's Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.6 Green's Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.7 Self-Adjointness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.8 Orthogonality of Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . 66

5.9 Real Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.10 Unique Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.11 Rayleigh Quotient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.12 More Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6 Variational (V) and Minimization (M) Formulations

97

Ordinary Differential Equations

Igor Yanovsky, 2005

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7 Euler-Lagrange Equations

103

7.1 Rudin-Osher-Fatemi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7.1.1 Gradient Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7.2 Chan-Vese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

8 Integral Equations

110

8.1 Relations Between Differential and Integral Equations . . . . . . . . . . 110

8.2 Green's Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

9 Miscellaneous

119

10 Dominant Balance

124

11 Perturbation Theory

125

Ordinary Differential Equations

Igor Yanovsky, 2005

5

1 Preliminaries

Cauchy-Peano.

du dt

=

f (t, u)

u(t0) = u0

t0 t t1

(1.1)

f (t, u) is continuous in the rectangle R = {(t, u) : t0 t t0 + a, |u - u0| b}.

M

=

max |f (t, u)|,

R

and

=

min(a,

b M

).

Then u(t)

with

continuous first derivative

s.t. it satisfies (1.1) for t0 t t0 + .

Local Existence via Picard Iteration.

f (t, u) is continuous in the rectangle R = {(t, u) : t0 t t0 + a, |u - u0| b}.

Assume f is Lipschitz in u on R.

|f (t, u) - f (t, v)| L|u - v|

M

= max |f (t, u)|, and

R

=

min(a,

b M

).

Then

a

unique

u(t),

with

u,

du dt

continuous

on [t0, t0 + ], (0, ] s.t. it satisfies (1.1) for t0 t t0 + .

Power Series.

du = f (t, u)

dt

u(0) = u0

u(t)

=

j=0

1 j!

dj u dtj

(0)tj

i.e.

d2u dt2

(0)

=

(ft

+

fuf

)|0

Fixed Point Iteration.

|xn - x| kn|x0 - x|

k ................
................

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