Math 309F
Final Exam
Page 1 of 9
Name:
Problem
Points
1
8
2
6
3
8
4
6
5
5
6
6
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6
8
7
Total
Score
52
Instructions
If you need more space to solve a problem, use the back of the page, and indicate that you have
done so.
Read each question carefully, and
Name:
Math 309 Quiz 2
a. Find the general solution of x =
4 5
5 4
x, expressed in terms of real-valued functions.
Solution. Find the eigenvalues of the matrix
det
4 5
5
4
4 5
5 4
:
= (4 )2 + 5 = 0
gives = 4 5i.
Next, nd an eigenvector for one of these eig
7.1 #16.
Solution. We want to show that x = x1 x2 , y = y1 y2 is a solution to the homogeneous
linear system
x = p11 (t)x + p12 (t)y
y = p21 (t)x + p22 (t)y,
i.e. we must show that
x1 x2 = p11 (t)(x1 x2 ) + p12 (t)(y1 y2 )
(1)
y1 y2 = p21 (t)(x1 x2 ) + p2
Name:
Math 309 Quiz 2
a. Find the general solution of x =
4 5
5 4
x, expressed in terms of real-valued functions.
b. Draw a direction eld for the system in part (a). You only need to draw two vectors, at the
points (0, 1) and (1, 0). Write down what each
Name:
Math 309 Quiz 1
a. Find the eigenvalues of the matrix A =
4
6
1 1
. For each eigenvalue, nd an eigen-
vector.
b. Use part (a) to write down two linearly independent solutions to the system of dierential
equations x = Ax.
1
Name:
Math 309 Quiz 1
4
6
1 1
a. Find the eigenvalues of the matrix A =
. For each eigenvalue, nd an eigen-
vector.
Solution. The characteristic polynomial of A is
4
6
1 1
det(A I) = det
= (4 )(1 ) + 6 = 2 3 + 2 = ( 1)( 2),
which has roots = 1, 2. These
Math 309 Homework 6
Due Wednesday, May 22.
Textbook problems:
10.2: 14b, 18b
10.3: 17*
10.4: 2, 3, 17, 18, 33
* In this problem youre asked to show Parsevals equation formally. This means to
show the formula symbolically without worrying about converge
Math 309 Homework 1
Due Wednesday, April 10.
Textbook problems:
7.1: 2, 16.
7.2: 1, 10, 15, 23.
7.3: 1, 31 (hint: how does A being singular relate to solving the system Ax = 0?).
1. Say whether each of the following systems of ODEs is linear or nonline
Math 309 Homework 7
Due Monday, June 3.
Textbook problems:
10.5: 12*, 18 (see Table 10.5.1), 22
10.6: 6, 9ab, 15a
10.7: 1abc, 5abc, 16abc*
* Recall that a steady-state solution u(x, t) of the heat equation is one with ut = 0.
* Do this problem after #1
Math 309 Homework 4
Due Friday, May 3.
Textbook problems:
7.9: 4 (youll need the result of #1 for this)
1. You probably saw the method of integrating factors in 307 for solving linear rst-order
ODEs, i.e. those of the form
y + p(t)y = q(t).
(1)
If we set
Math 309 Homework 3
Due Wednesday, April 24.
Textbook problems:
7.5: 29*, 31*
7.6: 1, 6, 7
* Notice that you are not solving any dierential equations here, only rewriting a secondorder equation as a system and computing a characteristic polynomial. For
Math 309 Homework 3
1. (a) Suppose A is an invertible constant matrix. Show that the only solution to the equation
x = Ax with x being a constant vector is x = 0. (A constant solution to an ODE is
called an equilibrium.)
Solution. Suppose x is a constant
Math 309 Homework 2
Due Wednesday, April 17.
Textbook problems:
7.3: 11, 14, 17, 18
7.5: 1, 2, 7, 15
Youre welcome to use a computer to help with the phase plane problems (e.g. the PPLANE
applet linked from the course webpage), although your solutions s
Math 309 Homework 2 solutions
4
2 40 40
1. The vector 2 is an eigenvector for the matrix A = 1 14 16 . What is its
1
2 3 6
eigenvalue? (Theres no need to calculate any determinants or nd any roots here!)
Solution. Saying that v is an eigenvector for A mea