Math 2233
Homework Set 3
1. Solve the following initial value problem.
y y = 2xe2x ; y(1) = 0 .
This is a first order linear ODE with p (x) = 1 and g(x) = 2xe2x . So
x
x
(x) = exp
p(s)ds = exp
ds =
Math 2233
Homework Set 9
1. Compute the Laplace transform of the following functions.
(a) f (t) = t
Let f (t) = t.
Z
tet dt
L[f ] =
0
Integrating by parts, with
u
du
dv
v
= t
= dt
= est dt
= 1s est
w
Math 2233
Homework Set 1
1. Determine the order of the following differential equations, whether or not the equations are linear and
whether the differential equations are ODEs (ordinary differential
Math 2233
Homework Set 2
1. Solve the following differential equation using Separation of Variables.
dy
= xey
dx
We can explicitly separate the x-dependence from the y-dependence in this equation by
Math 2233
Homework Set 7
1. Combine each of the following power series expressions into a single power series.
(a)
(n + 1)(x 1) 1 + n(x 1)
n
n=1
n
n=0
First we shift the summation index of the first
3. (25 points) Suppose a brine containing 0.2 kg of salt per liter runs into a tank initially field with 500L of
water containing 5 kg of salt. The brine enters the tank at a rate of 5L/min. The mixtu
Exam 1 MATH 2233 (Differential Equations) Spring 2013
Name(Last,First) ng Section
You must show all steps in your calculations or give explanations of your answers in order to get full credit.
You may
Math 2233
Homework Set 6
1. Solve the following Euler-type equations.
(a) x y + xy + y = 0
2
Since the differential equation is of Euler-type we expect solutions of the fomr y(x) = xr . Substituting
Math 2233
Homework Set 4
1. Find an integrating factor for each of the following differential equations and obtain the general solution.
(a) y + (y x)y = 0
Suppose (x, y) is an integrating factor for
Math 2233
Homework Set 8
1. Determine the lower bound for the radius of convergence of series solutions about each given point xo .
(a)
y + 4y + 6xy = 0 , x0 = 0
Since the coefficient functions
p (x
LECTURE 32
Systems of First Order Linear ODEs
In this lecture we will consider rst order ordinary dierential equations in which more than one unknown
function occurs. Let begin with a denition
s
Defin
LECTURE 30
Application of Laplace Transforms to Initial Value Problems
Consider the following initial value problem.
(30.1)
y 00
y0
2y = 0 ;
y 0 (0) = 0 :
y(0) = 1 ;
One method of solving this initial
LECTURE 31
Laplace Transforms and Piecewise Continuous Functions
We have seen how one can use Laplace transform methods to solve 2nd order linear Di E with constant
s
coe cients, and have even pointed
LECTURE 27
Singular Points and the Convergence of Series Solutions
As it stands our method of nding power series solutions to dierential equations of the form
(27.1)
y + p(x)y + q(x)y = 0
is purely fo
LECTURE 25
Solving Dierential Equations Using Power Series
We are now going to employ power series to nd solutions to dierential equations of the form
(25.1)
y + p(x)y + q(x)y = 0
where the functions
LECTURE 26
Dierential Equations with Polynomial Coecients
In the last lecture we considered a number of examples of dierential equations of the form
(26.1)
P (x)y + Q(x)y + R(x)y = 0
and looked for so
LECTURE 28
Series Solutions about Regular Singular Points
Recall x1 is a singular point for a dierential equation
y 00 + p (x) y 0 + q (x) y = 0
if
lim p (x) does not exist
x!x1
or
lim q (x) does not
LECTURE 29
The Laplace Transform
Suppose f : R ! R is a nice (to be qualied latter) function of x. The Laplace transform L[f ] of f is
the function from R to R dened by
(29.1)
Z
L[f ](s) =
1
e
sx
f (x
Homework - January 19
Instructions: Carefully written solutions to these problems are to be comnpleted by
Friday, January 19, and included in your homework notebook. Show all steps and
state what rule
Math 2233, Exanl I, Sept. 16, 2008 Name:
Score:
Part 1: Multiple choice. Each question is worth 4 points.
1. ( e ) Consider the autonomous differential equation y 2 y). The graph of f (y) is g
2
x3 =
2
1
x
1 + 3x
N
y
y 2x3
M
x3 =
=
=
N
M
0 = y d 1 + 3 x + x d y 2 x3 ) b (
.
1 + 2 x = ) x( y
C
sdleiy y rof noitaler siht gnivloS
. tnatsnoc yrartibra na C htiw , C = y + y 2x
snoitaler
ciarbe
Homework 2 solutions, 6.4,6.5,6.6,6.7
Section 6.8
2. Use the chain rule: 2e2x .
4. Also use the chain rule.
20. Use the product rule: x2 ex + 2xex .
26. Use the quotient rule.
33. Use the quotient rul
Homework 1, 6.1,6.2,6.3,
Due Friday, 1-26-18
Math 2133 Tech Calc II
When you start a new section, start a new page.
The total number of pages that you turn in
should be between 4 and 6 pages. Make sur
Homework 2 solutions, 6.4,6.5,6.6,6.7
Section 6.4
1. arctan(1) = /4, because tan /4 = 1. Notice that tan 5/4 = 1 as well, but arctan
asks for an angle
between /2 and /2, andso the answer
is /4.
4. a
MATH 2233
Short Test January 19
Students who correctly do 8 of the 10 problems will recieve 50 points. Those who do not get at
least 8 of 10 problems correct will be required to meet with the instruct