Math 4233
Solution to Homework Set 1
1. Find the inverses of the following matrices:
1
2
(a)
4
3
For invertible 2 2 matrices the following identity (easily derived from the cofactor expression for
A1 ) holds
1
d b
ab
=
c a
cd
ad bc
and so
1
4
3
1
3 4
11
Math 4233 Solutions to Homework Set 2
1. For each of the following systems find the fundamental (independent) solutions. (a)
dx dt
=
3 2
-2 -2
x
3- 2 eigenvector for = 2 : 0 = det N ullSp 3-2 2 -2 -2 - -2 -2 - 2 = 2 - - 2 = ( - 2) ( + 1) = = 2, 1
= N ull
Math 4233 Solutions to Homework Set 3
1. For each of the following inhomogeneous linear systems nd the general solution. (a) x = 2 3 1 2 x+ et t
First we solve the homogenous linear system. x = 2 3 1 2 x 2 3 1 2 ,
The eigenvalues and eigenvectors of the
Math 4233 Solutions to Homework Set 7
Before applying various numerical methods, let's write down the exact solution of x = 2x - 3t x (0) = 1 This is a first order, linear, non-homogeneous ODE with an initial condition. Such ODE/BVPs can be solved exactly
Math 4233 Homework Set 3
1. For each of the following inhomogeneous linear systems nd the general solution. (a) x = 2 3 1 4 2 3 1 2 1 1 1 2 x+ et t 2 1 1 1 et
(b) x =
x+
(c) x =
x+
et
2. Suppose that A is a constant n n matrix. Dene 1 1 exp (At) = I + At
Math 4233 Homework Set 1
1. Let x = x1 x2 and y = y1 y2 be two complex vectors. Show that (x, y) = (y, x) 2. If A is a hermitian 2 2 matrix and x and y are 2-dimensional complex vectors as above, show that (Ax, y) = (x, Ay) 2. Find the inverses of the fol
Math 4233 Homework Set 4
1. For each of the following systems carry out the following steps. (i) Identify the critical points. (ii) For each critical point c, identify the corresponding linear system. Write down the general solution of these linear system
Math 4233 Homework Set 7
1. (a) Use the Euler method with a step size of 0.1 to determine an approximate value of the solution of (1) x = 2x - 3t , x (0) = 1 at t = 0.4. (b) Repeat using a step-size of 0.01. 2. Use the Huen method (also known as the secon
Math 4233
SOLUTIONS TO SECOND EXAM
Thursday, July 19, 2012
1. Find the solution of the following heat conduction problem. Explain the steps you take in solving this
problem in as much detail as possible.
(1a)
4ut uxx = 0
(1b)
u (0, t) = 0
(1c)
u (2, t) =
Math 4233
SOLUTIONS TO FIRST EXAM
June 28, 2012
1.
(a) Find the eigenvectors and eigenvalues of the following matrix
1
2
2
1
To nd the eigenvalues, we determine the roots of the characteristic polynomial of the matrix:
0
=
2
1
p () := det
1
2
= 2 2 1 = 2
Math 4233 Homework Set 6
1. Determine if the following ODEs are of the Sturm-Liouville type d dy p (x) - q (x) y + r (x) y = 0 , p (x) > 0 , r (x) > 0 dx dx and if so identify the functions p (x) , q (x) and r (x). (a) 1 + x2 y - 2xy + l (l + 1) y = 0 (b)