Some Useful Facts About Integrals of Special Functions To Remember
1. Suppose that f is an integrable odd function. Then for any real number
a2R
Za
f (x) dx = 0:
a
2. Suppose that f is an integrable even function. Then for any real number
a2R
Za
Za
f (x)
Mathematics 81
Spring, 2015
Name: _
Homework Set 8
Due Monday, 3/9/15
1.
Find the interval of convergence for the power series:
k!( x 1)
k
k=0
2.
n ( x 3)
Find the interval of convergence for the power series: 3
n +1
n =1
3.
Rewrite the given series as a
Mathematics 81
Spring, 2015
Name: _
Homework Set 9
Due Monday, 3/16/15
1.
Find the power series solutions (in function form) for the differential equation:
y + y = 0
Then solve the equation by non-series techniques to verify your answer.
2.
Find the power
lm/A0
)4] only (,6 x: 50 59/1;5 (omfhgw owj 6W CZ
0-H)
m 1Q : .Dfrf Zgglglm _5_ 512$:ng )
W0 (rm-03H ; 333%.); a
Q (nr0(x~)w n39! >52, wtgg: a j ( 0%
- ' "w M W MW? :3 , "N , , WV?
aRD-n (M 03 h((,3) 5, h 4 n. Y 7"
i i be
3 ' .3 0" n
.1 ll? h L Did-m.
Mathematics 81
Spring, 2015
Name: _
Homework Set 12
Due Monday, 4/20/15
t
1.
Solve the integral equation: f ( t ) = cost + e f ( t ) d
0
2.
3.
4.
5.
Find the Laplace transform of the triangular wave (#52 in 7.4).
s 3
Find L1 ln
.
s +1
Hint: Consider L c
Mathematics 81
Spring, 2015
Name: _
Homework Set 11
Due Monday, 4/13/15
1.
3
if 0 t < 4
Find the Laplace transform of f (t ) , where f (t ) = 5 if 4 t < 6 , by writing f (t ) in
t
e
if t 6
terms of U(t a) , where U is the Heaviside function. Compare with
Problem 1 (11.2 #17). Determine whether the given boundary value problem is self-adjoint:
(1 + x2 )y + 2xy + y = (1 + x2 )y
y(0) y (1) = 0, y (0) + 2y(1) = 0
Solution. First notice that we can rewrite the dierential equation in the form:
[(1 + x2 )y ] + y
Throughout these notes we will frequently refer to the Sturm-Liouville problem:
[p(x)y ] + q(x)y = r(x)y,
0<x<1
1 y(0) + y (0) = 0,
1 y(1) + 2 y (1) = 0
(1)
the function r(x) is called the weight function of the Sturm-Liouville problem. The boundary condi
Math 146C - Ordinary and Partial Dierential Equations III - Spring 2011
April 28, 2011
Practice Midterm
Name:
Problem
Score
1
/25
2
/25
3
/25
4
/25
Score
/100
1
2
Problem 1 (25 points). Find all the eigenvalues and eigenfunctions for the boundary value pr
Math 146C ~ Ordinary and Partial Differential Equations III - Spring 2011
April 28, 2011
Practice Midterm
Name: 50:515m H 2
Problem 1 (25 points}. Find the Fourier series for the function
x) $252, ~7r So; SW.
'7F is even 22> bnao
DLuJT
7T 2 l
if: (M
Mathematics 81
Spring, 2015
Name: _
Homework Set 5
Due Tuesday, 2/17/15
1.
Solve the homogeneous equation: 2 y + 9 y + 12 y + 5y = 0
2.
Solve the homogeneous equation: 2y( 4 ) + 3y( 3) + 2y( 2 ) + 6 y 4y = 0
3.
Find the general solution of y + 6 y + y 34y
Mathematics 81
Spring, 2015
Name: _
Homework Set 2
Due Monday, 1/26/15
Please work problems on separate paper and attach this page to the front. Solutions must be
complete and in order to receive credit.
(
)
dy
+ cos 3 x y = 1
dx
1.
cos 2 x sin x
2.
dy 1
Mathematics 81
Spring, 2015
Name: _
Homework Set 3
Due Friday, 1/30/15
Please work problems on separate paper and attach this page to the front. Solutions must be
complete and in order to receive credit.
Solve each differential equation or initial value p