Rainwater pours off of a roof into a cylindrical
barrel. When it stops raining, water leaks out of
the barrel at a rate proportional to the square
root of the depth of the water at that time. The
water level drops from 49 inches to 25 inches in 2
hours.
a
Math 270
Answers to the Sample Test for Ch. 2
(not guaranteed correct!)
1.
The system is inconsistent.
2.
x = 3a, y = a, z = 2a [Other, equally correct forms of the answer exist.]
3.
x = 2, y = 7, z = 9
4.
300
5.
1
2
6.
Counterexample: A = B =
0 4 2
Math 270
Sample Test #2
March 2014
This exam is closed book. You must show all of your work for full credit. Be sure that you
explain what you are doing. No credit can be given for work that is unclear. Please put no
more than one problem on each side of
Math 270 Another Example for section 7.4
Consider the system of differential equations:
This can be written as: X = AX where A =
3
2
8
x = 3x z
y = 2x + 2y + z
z = 8x 3z
0 1
2 1
0 3
1
and X(0) = 2
8
So, lets find the eigenvalues.
3
0
1
2  2

Section 4.1, Page 182, Problem #10:
Determine the eigenvalues of the given matrix and find the corresponding
eigenvectors.
1 2 3
1 4 3 So we need to solve:
A =
1 2 1
Expanding around the first column gives:
1
2
3
1  4
3
1
2 +1
=
0
 4
(  1)
Example: Consider the subspace of
4
spanned by the vectors
v1 = <1, 0, 0, 1>, v2 = <1, 2, 1, 0>, v3 = <1, 1, 1, 0>, and v4 = <1, 1, 0, 1>.
a)
b)
c)
Find an orthogonal basis for this subspace.
Find an orthonormal basis for this subspace.
What is the dimen
Page 137 #11
n
n
Let U be a subspace of . Let U be the set of all elements of
that are orthogonal to every
element of U; that is, v is in U if v u = 0 for every element u of U. Show that U is a
n
subspace of . (The subspace U is called the orthogonal comp
Math 270
Sample Test for Ch. 2
1  3: Solve the following systems of linear equations using Gaussian
elimination. If you want full credit, you must show your work clearly and give the
full solution.
1.
x1
2x1
x1
+

x2
x2
4x2
x2
+
+
+
x3
x3
4x3
x3
+
+
+
+
Simplifying complex eigenvalues and eigenvectors
If your system of differential equations has real coefficients and real
variables, it is not particularly satisfying to have your answer involve
complex eigenvalues and eigenvectors. Fortunately, your answe
A yam is put into a 200C oven. The rate of change of its temperature is directly proportional to
the difference of its temperature and the temperature of the oven.
a)
Formulate the differential equation which describes this situation.
b)
If the yam is at
METHODS OF INTEGRATION
Keep this sheet handy for the rest of your life.
1)
To integrate xneax, xnsin(ax), or xncos(ax) where n is a positive integer and a is a
constant, integrate by parts n times, differentiating the power of x and integrating the
second
Consider the linear differential operator:
n
n1
L = a0(x)D + a1(x)D + . . . +an1(x)D + an(x)
The homogeneous differential equation Ly = 0 has n solutions which
u1(x)
u2(x)
.
we can write as
u=
.
.
un(x)
Assume that the solutions of the nonhomogen
Math 270
Sample Test #4
2014
Some more answers
(Problems with answers have their numbers in green.)
Remember, you can always check to see if your solution to a differential
equation is correct by putting it into the equation and seeing if it works.
1.
Sol
What is a Linear Space/Vector Space?
The terms linear space and vector space mean the same thing and can be used
interchangeably. I have used the term linear space in the discussion below
because I prefer it, but that is a personal preference.
To start wi
Numerical Methods
This handout introduces numerical methods which can be used to approximate the solution to
an initial value problem for a first order differential equation. In all of the methods the following
notation will be used:
The generic initial v
Example: Page 155 #12:
Verify that the set of all polynomials of degree <3, with domain restricted to [0,1],
is a vector space of dimension 3. Defining a scalar product on this space as
1
<f,g> = f(x) g(x) dx
0
find an orthogonal basis for this space. Sug
Example of a linear transformation proof
1
If V is the space of all continuous functions on [0,1] and if Tf = f(x) dx
for f in
0
1
V, show that T is a linear transformation from V into R .
1
f(x) dx is defined and is a Real
If f(x) is continuous on [0,1]
Section 1.7, page 66, problem #6
A tank initially contains 20L of water. A solution containing 1 g/L of chemical
flows into the tank at a rate of 3 L/min, and the mixture flows out at a rate of 2
L/min.
a)
b)
Set up and solve the IVP for A(t), the amount
Section 4.1 Page 182 #15
Show that a square matrix is singular if and only if 0 is an eigenvalue.
Let A be a square matrix.
Recall that A is singular detA = 0.
2 directions to prove:
= 0 A is singular
A is singular = 0
Given: = 0
We know that det(I A)
Section 1.8, page 76, problem #20
2x(y + 2x) y = y(4x y)
First off, lets rewrite this as:
dy
y(4x  y)
= 2x(y + 2x)
dx
The function on the right hand side (RHS) is homogeneous of degree 0.
We make the substitution y = vx
y
y
=
=
vx + v
vx
vx + v
=
=
vx
=
Math 270 Example for section 7.4
Consider the system of differential equations:
x = 3x + 2y
y = 5x + y
3 2
This can be written as: X = AX where A =
So, lets find the eigenvalues.
5 1
3
2
= 0 Which gives 4 + 13 = 0 or = 2 3i
5 1
So, now lets find
Math 270
Spring 2014
Numerical Methods Homework Problem
Due March 26, 2014
This problem is to be done by you. You may request assistance from others, but the
work you turn in must be your own. In addition, you are expected to credit those
from whom you ob
Math 270
Sample Test #4 A Few Answers
Spring 2014
10.
Let T be the transformation from
= xf(x) for f in
3.)
P3 into P3 that is defined by the formula Tf(x)
P3. (P3 is the linear space of polynomials of degree less than
a)
Part a) came from a separate prob