A proof of Theorem 7.15 on page 81
Theorem 7.15 (slightly rephrased): Suppose f (t, x) is a function dened on a rectangle
R in the tx-plane. Suppose f and f /x are continuous on R, and also suppose that, for
some number M and all (t, x) in R,
f
(t, x) M .
Systems of Second Order Differential Equations
On pages 356-57, Section 8.4, Example 4.8, our book set up the dierential equations
describing the motion of two masses connected to three springs. (See the gure on page 356
and equations on pp. 356-57.) Each
The Method of Undetermined Coefficients
for Systems of Differential Equations
We can use methods like those in Sec. 4.5 to nd particular solutions of systems of the
form y = Ay + f (t), the same kind of equation as in Sec. 9.9, where y(t) is the column
ve
Math 356.01, last three homework assignments
Homework 33, due Monday, December 2
Section 13.2, page 644, do problems 14 and 16. Notice the boundary condition! Also
do the ones below. Make your answers as simple as possible.
1.
ut = 4uxx ,
0 x 1,
t > 0,
ux
Math 356.01
Homework 28, due Monday, November 11
Do these problems in the book, and also the one below.
10.5: 2, 6, 22 and 10.6: 31
In problem 22 you should nd that the energy can decrease but cannot increase.
Problem S. In our usual model for a spring, t
Math 356.01
Homework 19, 20, and 21
Homework 19, due Friday, Oct. 18
9.2: 33, 34, 41 and 9.3: 10, 11, 12, 14
When you look at the gures in Sec. 9.3, look at the blue arrows and try to ignore
the black arrows! Your sketches should convey information but do
Math 356.01
Homework 29, 30, 31, and 32
Homework 29, due Wednesday, November 13
Find the Fourier series in each of the rst three problems by calculating the integrals for the
coecients. If the function is odd or even, you can use that fact to simplify the