Answers to Selected Problems in Chapters 1 and 2
1. Problem Set 1.4.1
1
4)|f (x) f (0)| = x = x x. For f to satisfy a Lipschitz condition
on an interval containing the origin would require that 1/ x L
for some constant L on this interval, and that is not
Mathematics 273 Midterm Exam: Solutions February 10, 2011
The problems are weighted equally
1. Find the solutions of the initial-value problems on [0, ):
(a) y =
x2
y,
1+x3
y (0) = 1.
Solution: y = exp
(b) y =
x2
y 1 ,
1+x3
x s2
0 1+s3
1/3
ds = (1 + x3 )
Answers to Selected Problems in Chapters 2 and 3
1. PS 2.3.1
2) The solutions cos x, sin x, 1 have the Wronskian W = 1; again
it is constant since p1 (x) 0.
3) Solutions are 1, x, exp(x) and W = exp(x). The theorem in
question gives for the Wronskian W
Answers to Selected Problems in Chapter 3
1. PS 3.5.1
4) We can read o the characteristic polynomials. For (a) it is
2
( + 1) ( + 2) = 3 + 42 + 5 + 2 so a1 , a2 , a3 = 4, 5, 2. For (b) it
2
is + 4 ( 2) = 3 22 + 4 8 so a1 , a2 , a3 = 2, 4, 8.
5)
(a) The
Solutions to Problem Set 5
1. The rst three (with the problems restated):
(a) Consider the second-order, constant-coecient problem
Lu D2 u + a1 Du + a2 u = r(x)
with initial values u(0) = 0 , Du(0) = 1 . Dene the Laplace transform:
esx u(x) dx,
f (s) =
(1
Solutions for Problem Set 6
Answers for problems from Chapter 5 of the text
1. PS 5.1.1
1) Since 1 is a double root, we must have the relations = 1 21
and = 2 . With substitution w(z ) = z 1 ln z in the expression
1
Lw = z 2 w + xw + w you nd
Lw = z 1
2
1. Supplemetary Problems for 7th Problem Assignment
These problems concern the system of equations
w = B (z )w
(1)
where the coecient matrix B is analytic except for isolated singularities. Recall that a singular point is called Fuchsian if B has a pole o
Solutions to Problem Set 8
1. Supplementary problem: nd the volume evolution in phase space for the
following systems:
(a) The Lorenz system
x = (y x),
y = rx y xz,
z = xy bz,
where , r, b are positive constants.
(b) Eulers equations of rigid-body dynamic
Solutions to Problem Set 9
1. Supplementary Problems: these relate to the conditions (7.6) on the ow
(t, x) of a dynamical system.
(a) Solve the one-dimensional system
x = x2
explicitly for and verify condition (iii).
Solution: The ow is (t, x) = x/(1 + t
Answers to Selected Problems in Chapters 1 and 2
1. Problem Set 1.4.1
4) a) Two solutions are x(t) = t2 /4 and x(t) 0
b) If f (x) = x satised a Lipschitz condition on [0, a) there
would be positive constant L such that
x 0 < L x there,
2
or x > 1/L ; sin
Answers to Selected Problems in Chapter 1
1. Problem Set 1.2.1
1) The standard form is
x=
1 x2
.
1 + t2
5) The antiderivative is x(t) = (1/3)t3 et + C so, applying the
initial condition, one nds
x(t) = (1/3)t3 et + 3.
7) Applying formula (1.23) one nds
Math 273, Final Exam Solutions
1. Find the solution of the dierential equation y = y + ex that satises
the condition y (x) 0 as x +.
SOLUTION: y = yH + yP where yH = cex is a solution of the homogeneous equation and yP is a particular integral, which must
Solutions to Problem Set 8
PS 7.3.1
3) Substituting x1 = r cos , x2 = r sin leads to the system
r = rf (r), r = r.
The choices r = 1 and r = 2 give solutions with = 0 + t representing periodic solutions in the original variables. The singular
case r = 0
Solutions for Problem Set 7
PS 6.2.1
2) The equivalent integral equation is
x
y (x) = 1 +
y (s) ds
0
so successive approximations are y0 = 1 and
x
yn (x) = 1 +
0
yn1 (s) ds.
This gives
y1 (x) = 1+x, y2 (x) = 1+x+(1/2)x2 , y3 (x) = 1+x+(1/2)x2 +(1/6)x3 .
Solutions for Problem Set 6
Answers for problems from Chapter 5 of the text
1. PS 5.1.1
1) Since 1 is a double root, we must have the relations = 1 21
and = 2 . With substitution w(z ) = z 1 ln z in the expression
1
Lw = z 2 w + xw + w you nd
Lw = z 1
2
Solutions to Problem Set 5
Answers to Selected Problems in Chapter 4
2) The corresponding series are those for exp(z ), cosh z, sinh z respecitvely.
5) q (z ) = 2(1 + z )2 = 2 + 4z 6z 2 + . . . with ql = 2(l + 1)(1)l+1 .
From the recursion formula (4.9)
Supplemetary Problems for Chapter 2
1. Let u1 , u2 , . . . , un be continuous, real-valued functions on an interval
[a, b] of the real axis. Show that a necessary and sucient condition for
these functions to be linearly dependent there is that the Gram ma
A VIEW OF MATHEMATICS
Alain CONNES
Mathematics is the backbone of modern science and a remarkably efficient source
of new concepts and tools to understand the reality in which we participate.
It plays a basic role in the great new theories of physics of t