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
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
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
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
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
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
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.
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 o
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 possib
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 polyno
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