DIFFERENTIAL EQUATIONS
Eric A. Carlen
Professor of Mathematics
Rutgers University
January, 2014
2
Contents
1 INTRODUCTION TO DIFFERENTIAL EQUATIONS
1.1
1
What dierential equations are, and what it means to solve them . . . . . . . . . . . .
1
The descript
DIFFERENTIAL EQUATIONS
Eric A. Carlen
Professor of Mathematics
Rutgers University
January, 2014
2
Contents
1 INTRODUCTION TO DIFFERENTIAL EQUATIONS
1.1
What dierential equations are, and what it means to solve them . . . . . . . . . . . .
1.1.1
n
1
1
Some
MULTIVARIABLE CALCULUS, LINEAR
ALGEBRA AND DIFFERENTIAL EQUATIONS,
PART II
Eric A. Carlen
Professor of Mathematics
Rutgers University
January, 2012
2
Contents
1 WHAT DIFFERENTIAL EQUATIONS ARE
1.1
1
Dierential Equations . . . . . . . . . . . . . . . . . .
Prep Problems for Quiz 5 01:220:320:04 Spring 2014
You have 24 hours to allocate between leisure () and work ( = 24 ). Every hour you work you
receive a wage $. (If you work a fraction of an hour, you get the corresponding fraction of .) You buy
a consump
Practice Test for Test 2, Math 292, April 25, 2013
Eric A. Carlen1
Rutgers University
April 24, 2014
1. The dierential equation
t2 x (t) 3tx (t) + 4x(t) = 0
has polynomial coecients.
(a) Find one polynomial solution to this equation.
(b) Find the general
Practice Final Exam, Math 292, 2014
Eric A. Carlen1
Rutgers University
May 6, 2014
1. Find the general solution of
t3 x (t) + t2 x(t) x2 (t) = 2t4 .
2. Cinside the two equations
I (y )2 + y 2 = 1
and
II (y )2 y 2 = 1 .
One has a unique solution with y(t)
Practice Test I, Math 292 Spring 2014
Eric A. Carlen1
Rutgers University
February 27, 2014
1: (a) Find the general solution of
x (t) = x(t) + sin t
(0.1)
and nd the unique value of x(0) for which the solution is periodic.
(b) Let x1 (t) and x2 (t) be any
Solutions for Homework Assignment 7, Math 292, Spring 2014
Eric A. Carlen1
Rutgers University
April 29, 2014
1. Let
2
1 + (y )2
dx
y
I[y] =
1
Consider the problem of minimizing I[y] subject to y(1) = a and y(2) = b. Find the corresponding solution, or sol
Solutions for Homework Assignment 6, Math 292, Spring 2014
Eric A. Carlen1
Rutgers University
April 27, 2014
1. This problem concerns dAlemberts formula.
(a) Let L > 0. Let g(x) = x(L x) for 0 x L. Let g be doubly antisymmetric about x = 0
and x = L. Show
Homework Assignment 6, Math 292, Spring 2014
Eric A. Carlen1
Rutgers University
April 8, 2014
1. This problem concerns dAlemberts formula.
(a) Let L > 0. Let g(x) = x(L x) for 0 x L. Let g be doubly antisymmetric about x = 0
abd x = L. Show that on nL x (
Homework Assignment 3, Math 292, Spring 2014
Eric A. Carlen1
Rutgers University
February 11, 2014
1. Let v(x, y) be the vector eld dened on the right half-plane U = cfw_(x, y) : x > 0 by
v(x, y) =
x,
1
2y + x2 y 2
x2
.
The system corresponding to this v
Homework Assignment 3, Math 292, Spring 2014
Eric A. Carlen1
Rutgers University
March 3, 2014
1. Let v(x, y) be the vector eld dened on the right half-plane U = cfw_(x, y) : x > 0 by
v(x, y) =
x,
1
2y + x2 y 2
x2
.
The system corresponding to this vecto
Homework Assignment 7, Math 292, Spring 2014
Eric A. Carlen1
Rutgers University
April 21, 2014
1. Let
2
1 + y2
dx
y
I[y] =
1
Consider the problem of minimizing I[y] subject to y(1) = a and y(2) = b. Find the corresponding solution, or solutions, of the Eu
Homework Assignment 4, Math 292, Spring 2014
Eric A. Carlen1
Rutgers University
March 4, 2014
1. (10 points) Let A be the matrix A =
0 1
0
.
(a) Compute A2 , A3 and A4 Observe the patterns, and deduce a formula for Ak for all positive
integers k. (You wi
Homework Assignment 5, Math 292, Spring 2014
Eric A. Carlen1
Rutgers University
April 27, 2014
1. Find the exact solution of x (t) = v(x(t), t) and x(0) = 0 for
v(x, t) = 2t(1 + x) .
Stating from X0 = 0, compute the next 4 terms in the Picard iteration, n
Homework Assignment 5, Math 292, Spring 2014
Eric A. Carlen1
Rutgers University
March 14, 2014
1. Find the exact solution of x (t) = v(x(t), t) and x(0) = 0 for
v(x, t) = 2t(1 + x) .
Stating from X0 = 0, compute the next 4 terms in the Picard iteration, n
Solutions for the Exercises from Chapter 1
Eric A. Carlen1
Rutgers University
February 10, 2014
1.1 Find the general solution of the dierential equation
tx = 3x + t4
for t > 0. Find the corresponding ow transformation, and the particular solution with x(1
Homework Assignment 4, Math 292, Spring 2014
Eric A. Carlen1
Rutgers University
February 22, 2014
1. (10 points) Let A be the matrix A =
0 1
0
.
(a) Compute A2 , A3 and A4 Observe the patterns, and deduce a formula for Ak for all positive
integers k. (Yo
Solutions for Homework Assignment 2, Math 292, Spring 2014
Eric A. Carlen1
Rutgers University
February 11, 2014
1. Let v(x) = sin(x). For all 0 x , Find all solutions of
x (t) = v(x(t) ,
x(0) = x0 .
For which values of t is each solution dened?
Hint: It w
Challenge Problem Set 1, Math 292 Spring 2014
Eric A. Carlen1
Rutgers University
February 11, 2014
This challenge problem set concerns the matrix exponential function. For any n n matrix A
this is dened by
k
t k
A
etA =
k!
k=0
where A0 = I by denition. W
Challenge Problem Set 2, Math 292 Spring 2014
Eric A. Carlen1
Rutgers University
February 26, 2014
Let K =
3 2
2 6
. The object of this challenge problem set is to nd and study the solution of
x (t) = Kx(t) + f (t)
with x(0) = (1, 2)
and
and x (0) = (1, 1
Challenge Problem Set 4, Math 292 Spring 2014
Eric A. Carlen1
Rutgers University
April 23, 2014
This challenge problem set concerns the construction of Greens functions for Lu = f subject
to boundary conditions other than u(a) = u(b) = 0. As always, we ta
Note on the Brachistochrone Problem
Eric A. Carlen1
Rutgers University
April 21, 2013
1
introduction
Consider two points p and q in the x, y plane.
We suppose that the height of q is lower than that of p, so that the straight line segment from
p to q runs
Challenge Problem Set 5, Math 292 Spring 2014
Eric A. Carlen1
Rutgers University
April 30, 2014
This challenge problem set concerns nding particular solutions of higher order third order in
this case solutions of inhomogeneous linear equations.
Consider t
Homework Assignment 2, Math 292, Spring 2014
Eric A. Carlen1
Rutgers University
January 30, 2014
1. Let v(x) = sin(x). For all 0 x , Fnd all solutions of
x (t) = v(x(t) ,
x(0) = x0 .
For which values of t is each solution dened?
Hint: It will probably hel