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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 ....
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