Assignment 420-13-1
To be returned September 5, 2013
Main part
In the problems 1 3 plot the bifurcation diagrams
1. (7) x = ax + 3
2. (9) x = x2 ax
3. (9) x = x3 x + a
In the next three problems, plot the integral curves, then solve the
equations, and com
Math 4200, Problem set 4
Solutions
October 3, 2013
Problem 1. i2013
Solution. It follows from the fact i4 = 1 that i2013 = i.
Problem 2.
1+i
1i
2013
Solution. After simplication we see that
Problem 3.
= i, which reduces to Problem 1.
2013
1+3i
1 3i
Soluti
Math 4200, Problem set 5
Solutions
October 10, 2013
Problem 1. x + 2x + x = 0
Solution. The characteristic equation is ( + 1)2 = 0 hence the general solution is given by
x(t) = ( + t)et .
Problem 2. x(3) 6 + 12x 8x = 0
x
Solution. The characteristic equat
Math 4200, Problem set 6
Solutions
October 22, 2013
Problem 1.
Solution. Without loss of generality we may assume that our cube is given by
ai ei R3
C=
0 ai 1 .
Let A be a linear endomorphism of R3 and dene i = Aei . Then clearly
a i i R 3
A( C ) =
0 ai 1
Math 4200, Problem set 7
Solutions
November 7, 2013
Problem 1. Solve x = Ax, x(0) = a, x(0) = b.
Solution.
a) A = ( 5 3 ), a = 0, b = e1 .
35
The matrix A has eigenvalues 2 and 8 with eigenvectors 1 =
respectively. In this way the matrix
1/ 1/2
2
B=
1/ 2
Math 4200, Prelim Exam
Solutions
October 30, 2013
Problem 1. Plot a bifurcation diagram for x = x4 x2 + a.
Solution.
2
1
0
1
2
1.0
0.5
0.0
0.5
1.0
Problem 2. For what values of a does the equation x = (sin5 t + a)x + cos t have a unique
2 periodic solutio
Prelim 4200-13-1
To be returned October 24, 2013
Main part
Notes and books are allowed; any collaboration is not allowed.
1. (10) Plot a bifurcation diagram for
x = x4 x2 + a.
2. (10) For what values of a the equation
x = (sin5 t + a)x + cos t
has a uniqu
Assignment 420-16-2
To be returned September 8, 2016
Main part
In Problems 13 find and plot the phase curves:
1. (3) x = y, y = x
2. (5) x = x + y, y = x + y
3. (7) x = x2 y 2 , y = 2xy
4. (5) For what values of the following equation has the uniqueness
p
Assignment 420-16-3
To be returned September 15, 2016
Main part
In the Problems 1 5 find and plot the phase curves.
1. ( 4) x = 2 x
2. (4 ) x = k 2 x
3. (6 ) x = x3 x
4. (6 ) x = sin x
5. ( 7+6+6) x = sin x + a, a = 21 , 1, 2
6. ( 7) Is the following vect
Math 4200, Problem set 3
Solutions
September 24, 2013
Problem 1. x = 2 x.
Solution. Following the general theory of conservative systems with one degree of freedom
let us dene the kinetic energy T and potential energy U as
T=
x
x2
,
2
2 d =
U =
x0
2 2
x
Math 4200, Problem set 2
Solutions
October 3, 2013
Problem 1. x = y , y = x.
Solution. This system is equivalent to the second order equation x = x. A particular
solution is x(t) = a cos t in which case y (t) = a sin t. This gives all the solutions of the
Assignment 420-13-2
To be returned September 12, 2013
Main part
In Problems 13 nd and plot the phase curves:
1. (3) x = y, y = x
2. (5) x = x + y, y = x + y
3. (7) x = x2 y 2 , y = 2xy
4. (5) For what values of the following equation has the uniqueness
pr
Assignment 420-13-3
To be returned September 19, 2013
Main part
In the Problems 1
5 nd and plot the phase curves.
1. ( 4) x = 2 x
2. (4 ) x = k 2 x
3. (6 ) x = x3 x
4. (6 ) x = sin x
1
5. ( 7+6+6) x = sin x + a, a = 2 , 1, 2
6. ( 7) Is the following vecto
Assignment 420-13-5
To be returned October 3, 2013
Main part
Solve
1. (5) x + 2x + x = 0
2. (5) x(3) 6 + 12x 8x = 0
x
3. (5) x + 2x + x = et
Find a periodic solution with the frequency , when exists, and plot
the graph of its amplitude as a function of :
Assignment 420-13-6
To be returned October 10, 2013
Main part
1. (6) Prove that the image of a cube under the action of a linear operator
is a parallelepiped, probably of lower dimension.
2. (8) Prove that the image of a ball under the action of a non-deg
Assignment 420-13-7
To be returned October 31, 2013
Main part
1. (7 each) Find the solutions and plot the orbits in the x-plane for the
following Cauchy problems: x = Ax, x(0) = a, x(0) = b
a)A =
53
, a=
35
0
0
= 0, b =
1
0
b)A =
53
, a = e1 , b = 0
35
c)
Assignment 4200-13-8
To be returned November 7, 2013
Main part
1. (12) Is the space of continuous functions on [0, 1] with the norm f =
1
|f (x)|dx complete?
0
2. (12) Prove that in assumption of the Banach Fixed Point Principle, the
distance between x an
Assignment 420-13-9
To be returned November 21, 2013
Main part
1. (10) (HSD 8.7) Plot the orbits of a system written in polar coordinates:
r = r r3 , = sin2 + a. Describe the global bifurcation at a = 1.
2. (12) (HSD 8.9) Plot the orbits of a system writt
Prelim 4200-13-2
To be returned November 14, 2013
Notes and books are allowed; any collaboration is not allowed.
1. (10) Find the image of the unit square [0, 1] [0, 1] under the action of
2 2
eA , A =
1 0
2. (10) Plot the orbits of a system x = Ax for
0
Assignment 420-16-5
To be returned September 29, 2016
Main part
Solve
1. (5) x(3) 6
x + 12x 8x = 0
2. (5) x + 2x + x = et
Find a periodic solution with the frequency , when exists, and plot
the graph of its amplitude as a function of :
3. (10) x + x = sin