Math 135: Ordinary Dierential Equations
Homework 1: Second Order Dierential Equations
Due on: Fri.,Jan. 15, 2016 - 9:00 AMi
Chapter 31 : Second order linear equations
Section 15, p. 91/92 : # 3 & # 11
Section 16, p. 94 : # 6
Section 17, p. 97/98 : # 1(h),
1. The Boundary Value Problems
Example 1
Find eigenvalues and eigenfunctions of the following equation
y 00 + y = 0
with boundary values y(0) = 0 and y(L) = 0.
Solution
The general solution is
y(x) = a cos( x) + b sin( x).
Then, apply the boundary values,
Math 135A, Fall 2015.
Homework 6, due Th., Nov. 26
1. A fair die is rolled twice. Compute the joint p.m.f. of X and Y , where X is the rst number rolled
and Y is the largest of the two numbers rolled.
2. Joint density of (X, Y ) is given by
f (x, y) = c(x
Math 135A, Fall 2015.
Homework 4, due Th., Nov. 5
1. Roll a pair of fair dice successively, and each time observe the sum rolled. What is the probability
of getting two 7s before getting six even numbers?
2. A bag contains 8 white, 4 black and 2 grey ball
Math 135A, Winter 2014.
Homework 7, due Tue., Dec. 3
1. Joint density of (X, Y ) is given by
f (x, y) = xex(y+1) ,
x, y > 0.
(a) Find the conditional density of Y given X = x. (b) Compute the density of Z = XY .
2. Assume that X1 , . . . , X10 are indepen
MAT 146
University of California
Spring 2014
Homework 1
due Friday April 11, 2014 in class
1. Stanley, Chapter 1.2
Suppose that the graph G has 15 vertices and that the number of closed
walks of length in G is 8 + 2 3 + 3 (1) + (6) + 5 for all 1. Let G
be
Math 135: Ordinary Dierential Equations
Homework 3: Laplace Transforms II
Due on: Fri. Jan. 29, 9:00- 9:50 AM
i
Problem 1
Use the Laplace transform technique to solve the following IVP:
y 6y + 9y = 0, y(0) = 0 and y (0) = 5,
Problem 2 Solve the following
Math 135: Ordinary Differential Equations
Homework 2: Laplace Transforms I
Due on: Fri., Jan 22, 2016, 9:00- 9:50 AM
Problem 1 Evaluate those of the following improper integrals that converge:
Z
1
(i)
dx
2
x +1
0
Z
1
(ii)
dx
x2
0
Problem 2 The Laplace i
Math 135: Ordinary Dierential Equations
Homework 4: Fourier Series
Due on: Fri. Feb. 5th, 9:00-9:50AM
Problem 1 Chapter 6: p. 256 # 2
Problem 2 Chapter 6: p. 256 # 4
Problem 3 Chapter 6: p.269 # 1
Problem 4 The function f (x) has period 2 and, over the ra
Math 135A, Fall 2015.
Homework 5, due Th., Nov. 12
1. Assume that weekly sales of diesel fuel at a gas station are X tons, where X is a random variable
with distribution function
c(1 x)4 0 < x < 1,
f (x) =
0
otherwise.
(a) Compute c. (b) Compute EX. (c) T
Math 135A, Fall 2015.
Homework 2, due Th., Oct. 15
1. Three tours, A, B, and C, are oered to a group of 100 tourists. It turns out that 28 tourists sign
for A, 26 for B, 16 for C, 12 for both A and B, 4 for both A and C, 6 for both B and C, and 2 for all
Math 135 Ordinary Differential Equations
Summer Session A 2013
Homework #3 Solutions
Instructions: Solve the following problems and be sure to show all of your work.
1. Discontinuous forcing terms. Consider the following piecewise-continuous function.
t
i
Math 135 Ordinary Differential Equations
Summer Session A 2013
Homework #2 Solutions
Instructions: Solve the following problems and be sure to show all of your work.
1. Laplace transforms. Using only its integral definition, compute the Laplace transform
Math 135 Ordinary Differential Equations
Summer Session A 2013
Homework #4 Solutions
Instructions: Solve the following problems and be sure to show all of your work.
1. Power series.
(a) Find a power series solution of the form
X
an xn
n=0
to the followin
Math 135 Ordinary Differential Equations
Summer Session A 2013
Homework #5 Solutions
Instructions: Solve the following problems and be sure to show all of your work.
For the following problems, let f and g be Riemann integrable functions on [a, b] and
b
Z
Math 135 Ordinary Differential Equations
Summer Session A 2013
Homework #1 Solutions
Instructions: Solve the following problems and be sure to show all of your work.
1. Finding Antiderivatives. Find the general solution to the differential equation:
ex
dy
Math 135A, Fall 2015.
Homework 3, due Th., Oct. 22
1. There are three bags: A (contains 2 white and 4 red balls), B (8 white, 4 red) and C (1 white 3
red). You select one ball at random from each bag, observe that exactly two are white, but forget
which b
Math 135A, Fall 2015.
Homework 1, due Th., Oct. 8
1. Fifteen married couples are at a dance lesson. Assume that (male-female) dance pairs are assigned
at random. (a) What is the number of possible assignments? (b) What is the probability that each
husband
Math 135: Ordinary Dierential Equations
Homework 5: Fourier Series & Boundary value problems
Due on: Friday, Februray 12, 2016, 9:00-9:50AM
Problem 1 The function f (x) has period 2 and it is dened by:
f (x) = eax for < x < .
Show that the Fourier expansi
Fourier Series
Problem 1
Given the Fourier series expansion of a 2-periodic function, f (x) dened in [, ]
f (x) =
a0
+
an cos(nx) + bn sin(nx),
2
n=1
show, using trigonometric identities (or otherwise), that the Fourier coecients are
given by:
1
a0 =
f (
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Math 135-2, Homework 1
Solutions
Problem 17.1
Find the general solution of each of the following equations:
(p) 16y 8y + y = 0
(q) y + 4y + 5y = 0
(r) y + 4y 5y = 0
(p)
0 = 16y 8y + y
y = erx
y = rerx
y = r2 erx
0 = 16r2 erx 8rerx + erx
= (16r2 8r + 1)erx