Math 196 - Quiz 5, October 2, 2008
Problem 1. In each of the following cases determine whether or not the subset W is a subspace of the
vector space V .
(1) W is the set of all vectors in V = R3 such that x1 + x2 = 0.
(2) W is the set of all vectors
Math 196 - Quiz 7, October 30, 2008
Problem 1. Compute the determinant of the following matrix in order to see if it is invertible:
4 7 10 2
1 0 2 0
0 2 0 0 .
1 7 1 2
Solution 1. If we switch two rows or columns of the matrix A then
and so we ha
Math 196 - Quiz 8, November 6, 2008
Problem 1. For the following matrix A, determine if it is diagonalizable and if so diagonalize it, i.e. nd
an invertible matrix P such that P 1 AP is a diagonal matrix.
1 1 1
A = 0 0 1 .
0 0 1
Solution 1. Note tha
Math 196 - Quiz 9, November 13, 2008
Problem 1. Find the general solution (on R) to the following dierential equation:
y (4) + y (2) = 0.
Solution 1. The associated polynomial to the above equation is P (x) = x4 x2 , which factors as P (x) =
(x2 + 1
Math 196 - Quiz 10, December 2, 2008
Problem 1. Find a particular solution on R to the dierential equation:
y + 2y + 3y = 4xex .
Solution 1. Using the method of undetermined coecients we rst notice that neither 4xex nor its derivative 4xex + 4ex are
Math 196 - Exam 1, September 23, 2008
Problem 1 (15 points). Give an explicit solution (or show that one does not exist) for the dierential
equation given by y 2 dxy = dx , under the initial conditions y (0) = 2, dx (0) = 1 .
Guidelines for the Method of Undetermined Coecients
Linear second-order nonhomogeneous constant coecient DE:
ay + by + cy = g(x)
sin x or cos x or
g(x) is a linear combination of products of (1), (2)
Math 2420 Final Exam Formulas
TRANSFORMS OF SOME FUNCTIONS
L[tn ] =
L[cos kt] =
L[sinh kt] =
L[sin kt] =
L[(t t0 )] = est0
LAPLACE TRANSFORMS - OPERATIONAL PROPERTIES
L[f (n) (t)] = sn F (s) sn1 f (0) sn2 f
Partial Fraction Decomposition of Rational Functions -
A. If the degree of P (x) Q(x), use long division to put in proper form.
B. Express Q(x) as a product of collected linear and irreducible quadratic factors.
ie. Q(x) = x2 (x 2)2 (x + 5)3 (x
Power Series Review
Taylor Series centered at a =
cn (x a)n = c0 + c1 (x a) + c2 (x a)2 + .
MacLaurin Series [Taylor series centered at a = 0] =
1. The series
converges when lim Sn =
cn xn exists. Otherwise
cn xn di
Math 196 - Quiz 4, September 25, 2008
Problem 1. Find all possible solutions to the following system of linear equations:
x + 2y z + 3w = 0
x 4y + 2 w = 0
2y z + w = 0
Solution 1. If we look at the associated matrix
to this homogeneou
Math 196 - Quiz 1, September 4, 2008
Problem 1. A Car accelerating at a constant rate goes from 0 to 60 miles per hour in 6 seconds, how far does the
car travel in this time?
Solution 1. If we let X (t), V (t), and A(t) be respectively the posit
-:>t Vi I
'"~ (.1-.1 .f(." ') l
1"\" II'\ "'0 ~
-f' l i-)
~ l~) ~ 1-
Homework 4, Section 3.3, Problem 32.
Show that the 2 2 matrix
is row equivalent to the 2 2 identity matrix provided that ad bc = 0.
Solution. If ad bc = 0 then we must have that at least one of a or c is not equal to 0. If a = 0 then
Partial Review of Differentiation and Integration
(c ) 0
( x ) nx n 1
(a ) a x ln x
(ln x )
(sin x) cos x
(cos x) sin x
( fg ) fg ' f g