Soluitons for Homework Problem Set 6 for Math 292 Spring 2012
Eric A. Carlen
Rutgers University
April 18, 2012
1: Suppose that the function x(t) has
x(1) = x(2) = x(7) = 0
and has no other zero in the interval 1 t 7. Suppose also that x(t) satises
x (t) +
Problem 1 (4pts per item). For each sentence below mark T for True or F for False. Fully
justify your answer. 1
There exist continuous functions p and q such that f (t) = t2 and g (t) = t3 are solutions
of y + p(t)y + q (t)y = 0.
If ej are eigenvectors of
RUTGERS UNIVERSITY
SPRING 2006
Due Date: Jan/30/06
Homework 2
Calculus VI - Math 292
Honors Section
Eduardo V. Teixeira
Department of Mathematics
Hill 440
Phone: 732-445-2473
E-mail: [email protected]
www.math.rutgers.edu/~eteixeir/
Problem 1 Show
RUTGERS UNIVERSITY
SPRING 2006
Homework 8
Due Date: April/10/06
Calculus VI - Math 292
Honors Section
Eduardo V. Teixeira
Department of Mathematics
Hill 440
Phone: 732-445-2473
E-mail: [email protected]
www.math.rutgers.edu/~eteixeir/
0
1
Problem
RUTGERS UNIVERSITY
SPRING 2006
Sample Problems for Exam 1
Calculus IV - Math 292
Honors Section
Eduardo V. Teixeira
Department of Mathematics
Hill 440
1
Phone: 732-445-2473
E-mail: [email protected]
www.math.rutgers.edu/~eteixeir/
First order dier
Challenge Problem Set 1, Math 292 Spring 2012
Eric A. Carlen1
Rutgers University
January 31, 2012
1
Introduction
This challenge problem set concerns the behavior of equations
x (t) = v (x(t)
(1.1)
where we do not have the Lipschitz condition
|v (x) v (y )
Challenge Problem Set 2, Math 292 Spring 2012
Eric A. Carlen1
Rutgers University
February 20, 2012
1
Introduction
This challenge problem set is about driven oscillatory systems.
32
. The object is to nd and study the solution of
Let K :=
26
x (t) = K x(t)
Challenge Problem Set 3, Math 292 Spring 2012
Eric A. Carlen1
Rutgers University
March 7, 2012
1
Introduction
This challenge problem set is about iteration and the contraction mapping theorem. In the exercises
that follow, let a > 0, and dene M to be the
Challenge Problem Set 4, Math 292 Spring 2012
Eric A. Carlen1
Rutgers University
March 21, 2012
1
Introduction
This challenge problem set is about using changes of variables based on symmetries to solve nonlinear systems.
Let U be the open positive quadra
Challenge Problem Set 5, Math 292 Spring 2012
Eric A. Carlen1
Rutgers University
April 5, 2012
1
Introduction
This challenge problem set is about solving the wave equation.
2
2
h(t, y ) = 2 h(t, y )
t2
y
(1)
with wave speed c = 1 subject to the boundary c
Challenge Problem Set 6, Math 292 Spring 2012
Eric A. Carlen1
Rutgers University
April 26, 2012
1
Introduction
This challenge problem set is about using the Ritz method to approximately compute eigenvalues
for ordinary dierential equations.
Consider the o
Solutions for Homework Problem Set 1
for Math 292 Spring 2012
Eric A. Carlen
Rutgers University
February 9, 2012
1. Find all solutions x(t) on t > 0 of
tx + x = 3t2 1 .
Find the solution with x(1) = 2.
SOLUTION: This can be written
(tx(t) = 3t2 1
so integ
Solutions for Homework Problem Set 2
for Math 292 Spring 2012
Eric A. Carlen
Rutgers University
February 20, 2012
1: Find the general solution of the Riccati equation
2
y = y + x3 y 2 + x5 ,
x
for x > 0.
SOLUTION: Since the coecients are powers of x, we l
Solutions for Homework Problem Set 3 for Math 292 Spring 2012
Eric A. Carlen
Rutgers University
March 1, 2012
1. Consider the system of dierential equations
x
= 2x + y
y
= y
(a) Find a 2 2 matrix A so that with x = (x, y ), this system can be written as
x
Homework Problem Set 4 for Math 292 Spring 2012
Due Monday, March 26
Eric A. Carlen
Rutgers University
March 4, 2012
1. Consider the system of dierential equations x = Ax where
5 3 2
A = 8 5 4 .
4
3
3
Find the general solution.
SOLUTION In a previous exer
Solutions for Problem Set 5 for Math 292 Spring 2012
Eric A. Carlen
Rutgers University
April 18, 2012
1. Consider the equation
y
2
y=0.
t2
(a) Let y1 (t); = t , and nd a values of that makes y1 a solution of the equation.
(b) Find the general solution to
332:221 Principles of Electrical Engineering I Fall 2004
Quiz 1
D
C
2
It is easy to see that all the circuit diagrams shown
on the right are electrically equivalent although
their physical layouts seem dierent. This is because the nodes C, E, and D are al