Solutions to Midterm Quiz #1
Math 119B
UC Davis, Spring 2012
Question 1
Dene the functional
/2
(q (t)2 q (t)2 ) dt
(q ) =
0
Find the function q where (q ) is minimal, subject to the boundary conditio
Solutions to homework assignment #1
Math 119B
UC Davis, Spring 2012
1. (a) Starting from the Hamiltonian H (p, q, t) = p2 + tq 4 , nd the associated Lagrangian, and
write the associated ODEs in both t
Solutions to review problems from HW #5
Math 119B (UC Davis, Spring 2012)
7. Use matrix exponentials, or any other method, to nd the solution (x(t), y (t) of the ODE
system
x = 2x + y,
y = x + 2y
sati
Solutions to nal exam (6/12/12)
Math 119B
UC Davis, Spring 2012
Question 1 (25 points)
Use matrix exponentials, or any other method, to nd the solution (x(t), y (t) of the ODE system
x = 2x y,
y = 2x
Solutions to midterm quiz #2
Math 119B
UC Davis, Spring 2012
Question 1
(a) Prove that if A, B are commuting square matrices (i.e., matrices which satisfy AB = BA) then
exp(A + B ) = exp(A) exp(B ).
S
Solutions to homework assignment #2
Math 119B
UC Davis, Spring 2012
1. In the computer game Angry Birds Space, let x(t) = (x(t), y (t) denote the position of
an angry bird at time t ying near a planet
Solutions to homework assignment #4
Math 119B
UC Davis, Spring 2012
1. A rope of length and uniform linear density hangs between two points (A/2, 0), (A/2, 0),
where we assume that A. Its shape y = q
Solutions to homework assignment #5
Math 119B
UC Davis, Spring 2012
1. Find the solution xn to the recurrence
xn+2 = xn+1 + 2xn
satisfying the initial conditions x1 = 5, x2 = 1.
Solution. As in the ex
Solutions to homework Assignment #6
Math 119B
UC Davis, Spring 2012
Problems
1. For each of the following maps acting on the interval [0, 1], sketch their graphs, nd their
xed points and determine for
Solutions to homework assignment #7
Math 119B
UC Davis, Spring 2012
1. A xed point x of an interval map T is called superstable if T (x ) = 0. Find the value of
0 < r 4 for which the logistic map Lr h
Solutions to homework assignment #8
Math 119B
UC Davis, Spring 2012
1. A child standing on a swing bends her knees up and down in a periodic motion. This causes a
slight change in the resonant frequen