Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: PROBLEM SET 5
DUE THURSDAY, FEBRUARY 22, 2007
b
Problem 1. Suppose that I = [a, b] is an interval. Let f 1 = a f (x) dx for
f C (I ; R) (i.e. f is a continuous realvalued function on I ), and let L1 (I ) denote
the completion of C (I ; R) wi
Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: PROBLEM SET 1
DUE THURSDAY, JANUARY 18, 2007
Problem 1. Suppose that (V, . ) is a normed vector space, A : Rn V is a linear
map. Let . 2 denote the standard norm on Rn . Show that there is C > 0 such
n
that Av C v 2 for all v Rn . (Hint: write
Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: PROBLEM SET 2
Suggested Solution
Problem 1. Recall that a map F between complex vector spaces is called dierentiable
(in the complex sense) at x if there is a complex linear map L such that F (x + y ) =
F (x) + Ly + R(x, y ) with lim y 0 R(x,y)
Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: PROBLEM SET 3
Suggested Solution
Problem 1. (Cf. Taylor I.1.3.) Let Mnn (C) denote the set of n n complex matrices.
Suppose A Mnn (C) is invertible. Using
det(A + tB ) = (det A) det(I + tA1 B )
show that
D det(A)B = (det A) Tr(A1 B ).
(Hint: yo
Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: PROBLEM SET 2
DUE THURSDAY, JANUARY 25, 2007
Problem 1. (Cf. Taylor, I.1.4) Recall that a map F between complex vector spaces
is called dierentiable (in the complex sense) at x if there is a complex linear map
L such that F (x + y ) = F (x) + L
Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: MIDTERM
DUE AT 4PM ON MONDAY, FEBRUARY 12, 2007
Problem 1. The purpose of this problem is to see how our results so far help us
solve rst order PDEs. Suppose O is an open subset of Rn , and consider a scalar
semilinear PDE on O, i.e. suppose th
Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: FINAL EXAM
THURSDAY, MARCH 22, 2007
There are three problems. Do all of them.
Problem 1. Consider the vector eld X (x, y ) = (y, x) = y x + x y in R2 .
(1) Find all integral curves of X .
(2) Show that the equation Xu = 0, u C 1 (R2 ), with ini
Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: PROBLEM SET 7
Suggested Solution
Problem 1. Use separation of variables to solve the Dirichlet problem for the Laplacian:
u = 0 in , u = f given, where = cfw_z R2 : a < z  < b, a, b > 0, is an annulus.
(Hint: remember that there are two line
Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: PROBLEM SET 5
Suggested Solution
b
Problem 1. Suppose that I = [a, b] is an interval. Let f 1 = a f (x) dx for f C (I ; R)
(i.e. f is a continuous realvalued function on I ), and let L1 (I ) denote the completion of
C (I ; R) with respect to
Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: PROBLEM SET 4
DUE WEDNESDAY, FEBRUARY 7, 2007
Problem 1. Suppose V is a nite dimensional vector space over F = R or F = C.
Its dual V is the vector space L(V, F) of linear maps from V to F. The elements
of V are called linear functionals on V .
Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: PROBLEM SET 6
Suggested Solution
Problem 1. For f, g C (S1 ), let
f g =
1
2
f ( )g ( ) d.
S1
(1) Show that f g C (S1 ) and f g f g , where f = supcfw_f () :
S1 .
(2) Show that f g = g f .
(3) Show that in fact f g f g 1 , where g 1 = 21 g (
Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: PROBLEM SET 8
Suggested Solution
Problem 1. (Taylor 3.3.15 ) Using Exercise 14, prove the Weierstrass approximation
theorem: Any f C ([a, b]) is a uniform limit of polynomials. (Hint: Extend f to u as
above, approximate u by p u, and expand thi
Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: FINAL EXAM
THURSDAY, MARCH 22, 2007
There are three problems. Do all of them.
Problem 1. Consider the vector eld X (x, y ) = (y, x) = y x + x y in R2 .
(1) Find all integral curves of X .
(2) Show that the equation Xu = 0, u C 1 (R2 ), with ini
Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: MIDTERM
DUE AT 4PM ON MONDAY, FEBRUARY 12, 2007
Problem 1. The purpose of this problem is to see how our results so far help
us solve rst order PDEs. Suppose O is an open subset of Rn , and consider a
scalar semilinear PDE on O, i.e. suppose th
Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: PROBLEM SET 4
Suggested Solution
Problem 1. Suppose V is a nite dimensional vector space over F = R or F = C. Its
dual V is the vector space L(V, F) of linear maps from V to F. The elements of V are
called linear functionals on V .
(1) Show tha
Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: PROBLEM SET 3
DUE THURSDAY, FEBRUARY 1, 2007
Problem 1. (Cf. Taylor I.1.3.) Let Mnn (C) denote the set of n n complex
matrices. Suppose A Mnn (C) is invertible. Using
det(A + tB ) = (det A) det(I + tA1 B )
show that
D det(A)B = (det A) Tr(A1 B
Topics in Differential Equations with Applications
MATH 174

Spring 2013
MATH 174A: PROBLEM SET 1
SUGGESTED SOLUTIONS
Problem 1. Suppose that (V, ) is a normed vector space, A : Rn V is a linear
map. Let 2 denote the standard norm on Rn . Show that there is C > 0 such that
Av C v 2 for all v Rn . (Hint: write v = n=1 aj ej , w