HOMEWORK #4 Solutions
1.[Strauss 5,p.41] The equation for the damped string is given by
utt c2 uxx + rut = 0,
with r > 0
The formula for energy is
1
E=
2
u2 + T u2 dx
t
x
where c2 = T /.
We look at the rst derivative of E
1
dE
=
dt
2
2ut utt + 2T ux uxt d
HOMEWORK #8 Solutions
6.4.1[S] Harmonic functions in the exterior of the disk cfw_r > a, bounded at innity, are
described by equation (8), p.174:
1
u(r, ) = A0 +
2
rn (An cos n + Bn sin n)
n=1
where r > a and [, ]. We also have the boundary condition
u(a,
2
MATH 428 - FORMULAE
6. Representation formulae:
6.1. Poissons formula. The Poisson Kernel on a disk
with radius a:
a2 r2
K (r, , a, ) = 2
.
a 2ar cos( ) + r2
Harmonic function on the disk with boundary value h():
2
1
K (r, , a, )h() d.
w(r, ) =
2 0
The
323 HOMEWORK I
Exercise 1: Find the general solution to:
x = t2 sin(t)x.
Exercise 2: Solve:
x = sin(t) cos(t)x + e 2 cos
1
2
(t)
,
given the initial value x(0) = e1/2 .
Exercise 3: Find the general solution to:
x = x + sin(t) + cos(2t).
Exercise 4: Find t
323 HOMEWORK II
Exercise 1: Find the solution to:
x + 2x + 5x = 0,
satisfying x(0) = 1, x(0) = 1. Compute x(5).
Exercise 2: Verify that 1 (t) = t2 is one solution to
t2 x + tx 4x = 0,
nd a second solution 2 (t) (independent of the rst), and
determine the
323 HOMEWORK III
Exercise 1: Let
11
A=
.
41
Find the general solution to x = Ax.
Exercise 2: Let
1 0 5
A = 0 8 0 .
5 0 1
Solve the equation x = Ax,
with the initial condition x(0) = (4, 6, 2).
Exercise 3: Let
0 1 0
A = 1 0 0 .
003
Solve the equation x = A
323 HOMEWORK IV
Exercise 1: Let
1 5 5
A=
.
213
Solve the equation x = Ax,
with the initial condition x(0) = (4, 3).
Exercise 2: Let
1
A=
9
38 10
4 16
.
Find the general solution to x = Ax + e3t
Exercise 3: Let
A=
0 1
10
1
.
1
.
Find the general solution t
323 HOMEWORK V
Exercise 1: Consider the equation:
2
x
x + y 2 25
=
.
y
xy 12
Locate all the xed points of the system, classify them, and
sketch them in the phase plane.
Exercise 2: Classify the non-zero equilibrium points of
the system:
x = x4 + y 8
.
y
323 HOMEWORK VI
Exercise 1: Classify the equilibrium points of the system:
x = x3 + y 5
.
y = x5 y
Exercise 2: Consider the equation:
2
x
x + ay
=
,
y
b(x + y )
where a and b are constants. The system has two equilibrium points when a = 0 and b = 0. Dete
3. Wave Equation:
MATH 428 - FORMULAE
3.1. One dimensional. utt c2 uxx = F (x, t),
u(x, 0) = f (x), ut (x, 0) = g (x)
x+ct
1. Characteristic equations
f (x + ct) + f (x ct)
1
u(x, t) =
+
g (s) ds
2
2c xct
1.1. Quasilinear. a(x, y, u)ux + b(x, y, u)uy = c
2
a local maximum. Since w does not have an interior positive maximum and it is zero
on the boundary, either w has an interior negative minimum or it is constantly equal
to 0. Assume that w has an interior negative minimum at q , then w(q ) = w(q ) < 0,
w
MATH 4280: Final
Thursday, May 17th
Answer the following 6 questions, 4 points each. Show all work. Closed book, no calculators. You are allowed to use the 428 formula sheet. Academic integrity on the part of
each student is presumed. Violations will be d
HOMEWORK #12 Solutions
9.1.1[S] We are looking for solutions for the wave equation
utt c2 u = 0
which have the form u(x, t) = f (k x ct) where k = (k1 , k2 , k3 ) is a xed vector and
x = (x, y, z ). We have
ut = cf (k x ct),
utt = c2 f (k x ct)
ux = k1 f
HOMEWORK #11 Solutions
1.Let D be a bounded domain in R2 and G1 (x, x0 ), G2 (x, x0 ) two Green functions
associated to this domain. Dene for i = 1, 2
Hi (x, x0 ) = Gi (x, x0 )
1
ln |x x0 |.
2
By property (iii), H1 and H2 are both harmonic,bounded and ha
HOMEWORK #10 Solutions
1. We want to solve the Neumann problem
u = 0, for 0 < r < 1, 0 2
ur (1, ) = A0 + 5 sin 3 2 cos
The solution of Laplaces equation on the disk is given by the Fourier series
rn (An cos n + Bn sin n)
u(r, ) = A0 +
n=1
Note that A0 =
HOMEWORK #8 Solutions
1. The solution of the diusion problem on a ring of radius 1 is given by the series
u(x, t) =
A0
+
2
2t
(An cos nx + Bn sin nx) en
n=1
where
A0
u(x, 0) =
+
2
An cos nx + Bn sin nx.
n=1
Given the formula we have for u(x, 0) and the un
HOMEWORK #7 Solutions
1.[Strauss 2, p.92] We have a wave equation on [0, l] with Neumann boundary condition
at the left: ux (0, t) = 0 and Dirichlet condition at the right: u(0, t) = 0. Using the method
of separation of variables, this reduces to solving
HOMEWORK #6 Solutions
1.[Strauss 10(a),p.53]Using formula (8) on page 49, we have
u(x, t) =
We make the change of variables p =
1
u(x, t) =
2
ep (x
e
(xy )2
4kt
y 2 dy.
x
y :
4kt
1
4kt
1
4ktp)2 dp =
2
ep (x
4ktp)2 dp.
2.[Strauss 15, p.53]The uniquen
HOMEWORK #5 Solutions
1.[Strauss 2,p.79] We want to solve the inhomogeneous initial value problem
utt = c2 uxx + eax , for < x < , t > 0
u(x, 0) = 0
ut (x, 0) = 0
The solution u(x, t) is given by formula (3), page 71, where and are both 0:
1
2c
u(x, t) =
HOMEWORK #3 Solutions
1.[Strauss 2,p.38] This an illustration of DAlemberts formula. We want to solve
utt = c2 uxx , for < x < , t 0
u(x, 0) = log(1 + x2 )
ut (x, 0) = 4 + x
(1)
We denote (x) = u(x, 0) and (x) = ut (x, 0). By Dalemberts formula, we have
HOMEWORK #2 Solutions
1.[Strauss 5p.10] We want to solve xux + yuy = 0. The characteristic curves are solutions
dy
y
of the equation dx = x , where y is a function of x. The solution of this ODE is given by
y = Cx, where C is an arbitrary constant. There
HOMEWORK #1 Solutions
1.We have the equation uxtt =exp(2x + 3t). Integrating with respect to t we get
1
uxt = e2x+3t + constant
3
but the constant may depend on x so actually we have uxt = 1 e2x+3t + f (x), where f is an
3
arbitrary function. Integrate ag
323 HOMEWORK VII
Exercise 1: Solve the equation xux + (x + y )uy = 1, with
the initial condition u(1, y ) = y . Describe the domain of
the solution.
Exercise 2: For x > 0, consider the equation xux + (x2 +
y )uy + (y/x x)u = 1. Solve the equation under th
323 HOMEWORK VIII
Exercise 1: Solve the equation:
utt uxx = xt,
in the domain < x < , t > 0, given the initial values u(x, 0) = 0,
ut (x, 0) = ex .
Exercise 2: Solve the equation:
utt 4uxx = ex + sin t,
in the domain < x < , t > 0, given the initial value