AnswerPSet_2010_01

AnswerPSet_2010_01 - Answers P-Set Number 1 18.385j/2.036j...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Answers P-Set Number 1, 18.385j/2.036j MIT (Fall 2010) Rodolfo R. Rosales (MIT, Math. Dept., room 2-337, Cambridge, MA 02139) September 21, 2010 Course TA: Jan Molacek, MIT, Math. Dept. room 2-331, Cambridge, MA 02139. Email: [email protected] Contents 1 Problem 2.5.4 - Strogatz (Infinitely many solutions). 2 1.1 Statement for problem 2.5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Answer for problem 2.5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Problem 2.5.5 - Strogatz (Example of non-uniqueness). 2 2.1 Statement for problem 2.5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Answer for problem 2.5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 Problem 2.5.6 - Strogatz (The leaky bucket). 3 3.1 Statement for problem 2.5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 Answer for problem 2.5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4 Problem 3.2.6 - Strogatz (Eliminating the cubic term). 5 4.1 Statement for problem 3.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.2 Answer for problem 3.2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5 Problem 3.2.7 - Strogatz (Eliminate high order terms). 7 5.1 Statement for problem 3.2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5.2 Answer for problem 3.2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 6 Problem 3.3.1 - Strogatz (Improved model of a laser). 8 6.1 Statement for problem 3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 6.2 Answer for problem 3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 7 Problem 3.3.2 - Strogatz (Maxwell-Bloch equations). 11 7.1 Statement for problem 3.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 7.2 Answer for problem 3.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 8 Problem 3.4.9 - Strogatz (1-D bifurcations). 14 8.1 Statement for problem 3.4.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 8.2 Answer for problem 3.4.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1 2 1 Problem 2.5.4 - Strogatz (Infinitely many solutions). 1.1 Statement for problem 2.5.4 (Infinitely many solutions with the same initial condition). Show that the initial value problem dx dt = x 1 / 3 , x (0) = 0 , (1.1.1) has an infinite number of solutions. Hint 1.1.1 Construct a solution that stays at x = 0 until some arbitrary time t , after which it takes off. 1.2 Answer for problem 2.5.4 It is clear that both x ( t ) ≡ 0 and (for t ≥ t ) x = 2 3 ( t- t ) 3 / 2 are solutions of the ODE, for any constant t ....
View Full Document

{[ snackBarMessage ]}

Page1 / 15

AnswerPSet_2010_01 - Answers P-Set Number 1 18.385j/2.036j...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online