Math 5310 Fall 2010
Homework set #1 SOLUTIONS
1. Problem 1.1:
(a)
(b)
(c)
dx
dt
+ xt = 0 is a linear ODE.
u
2 u
t x2 = (1 + t ) sin x is a linear
w
w
t + w x = 0 is a nonlinear PDE
PDE
2. Problem 1.5
(a) Try x (t) = et/2 as a candidate for solution to 1.1
Math 5310 Fall 2010
Homework set #2 SOLUTIONS
1. See Mathematica notebook
2. Algebra with oating point numbers
(a) Suppose our FP scheme keeps four digits and an exponent. For example, it can store a = 1.234
and b = 5.678 105 . It can add a + a to nd 2.46
Math 5310 Fall 2010
Homework set #3, solutions
Problem 1
Part (a)
The Laguerre polynomials are differentiable so we can use the Wronskian. In this case, the Wronskian matrix
is upper triangular with diagonal elements (1, 1, 1, 1). (You might ask yourself
Math 5310 Fall 2010
Homework set #3 SOLUTIONS
1. See Mathematica notebook
2. If the members of S are LD, then a nontrivial constant vector R N such that
N
i i (x) = 0
x [ a, b ] .
i =1
By hypothesis, i C ( N 1) [ a, b]. Differentiation is linear and the
Math 5310 Fall 2010
Homework #4 solutions
1. The binary operation
1
( f , g) =
f ( x ) g ( x ) dx
1
is clearly symmetric, ( f , g) = ( g, f ). It also bilinear,
1
( f + g, h ) =
1
1
f + g h dx =
f h dx +
1
1
g h d x = ( f , h ) + ( g, h )
1
and simila
Homework #6 solutions
Math 5311 Spr 2010
Problem 5.1.8
2
Part (a) The null space of K D , aka ker (K D ) , is the subspace of CD such that K D u = 0. To nd the kernel, then,
we need to solve the differential equation
Ku = au + bu = 0.
This is a second-ord