Ma 2a
(analytical track)
Fall 2011
PROBLEM SET 1
Due on Monday, October 3
[All references are to J. C. Robinson]
I. 1D autonomous equations (formula solution)
1. [Problem 8.7] Show that for k = 0 the solution of the IVP
x = kx x2 ,
x(0) = x0 ,
is
kekt x0
Ma 2a
(analytical track)
Fall 2011
PROBLEM SET 2
Due on Monday, October 10
I. Direction (slope) elds
1-2. Draw direction eld diagrams for the following equations:
(a) y = y 2 x, and
(b) y = y/(x2 + y 2 ).
(You can use a computer.)
II. Approximation of sol
Ma 2a
(analytical track)
Fall 2011
PROBLEM SET 3
Due on Monday, October 17
I. Methods of integration of 1st order ODEs
Problems 9.1(vi), 10.2, 10.4 (i)
(X1) Solve the equation
x2 y + 2xy y 3 = 0.
II. Applications of 1-st order ODEs
Problem 9.3
(X2) Find o
Ma 2a
(analytical track)
Fall 2011
PROBLEM SET 4
Due on Monday, October 24
I. Integration of 1st order equations
(X1) Solve the equation
x=
t
t2 x + x3
(Hint: s = t2 ).
(X2) Solve the initial value problem
y=
y3
,
1 2xy 2
y (0) = 1.
II. 2D autonomous syst
Ma 2a
(analytical track)
Fall 2011
PROBLEM SET 5
Due on Monday, November 7
I. Undetermined coecients
Problems 16.1(iv), 22.4(v)
II. Oscillations
Problems 13.4, 13. 8, 15.4
III. Linear systems with constant coecients
Problem 26.1(v)
(X1) Rewrite the n-th o
Ma 2a
(analytical track)
Fall 2011
PROBLEM SET 6
Due on Monday, November 14
I. Phase portraits of linear systems
Problem 28.6
Problem 31.1(i-ix)
[counts as 3 problems]
II. Linearization near critical point
Problem 32.1
III. Ecological models
Problem 33.1(
Ma 2a
(analytical track)
Fall 2011
PROBLEM SET 7
Due on Monday, November 21
I. Conservative systems
Problem 34.1(iii) (sketch the phase portrait by hand).
(X1) Find (approximately) the periods of small oscillations in Problem 34.1(iii)
II. Dissipative sys
Ma 2a
(analytical track)
Fall 2011
PROBLEM SET 8
Due on Friday, December 2
I. Boundary value problems
1. Consider the BVP
u + u = 0,
u (0) = u( ) + u ( ) = 0.
Find approximate value of the smallest eigenvalue 1 , and estimate n for n
1.
2-3. Find the eige